DYNAMICAL SAMPLING ON FINITE INDEX SETS
CARLOS CABRELLI, URSULA MOLTER, VICTORIA PATERNOSTRO,
AND FRIEDRICH PHILIPP
Abstract. Dynamical Sampling aims to sample spatio-temporal signals at fixed points
in space and at various times, thereby exploiting the knowledge on the time evolution of
the signal. In the corresponding mathematical model the samples of the signal f are of
the form hf, An fi i, where A is a bounded operator and the fi are fixed vectors. Here, we
characterize those normal operators A and finite systems (fi ) for which all signals f can
be stably recovered from the samples. The result is an interesting mixture of operator
theory, spectral theory, and complex analysis in the unit disk. We also consider general
bounded operators A and show in particular that it is necessary for stable recovery that
the spectrum of A∗ is contained in the closed unit disk and that the eigenvalues in the
open disk either fill it up completely or are discrete. It is shown that the first case is
necessary for non-redundant stable recovery.
1. Introduction
Given a system of vectors {fi }i∈I from some Hilbert space H and a linear operator
A, we analyze the collection of iterates {An fi : i ∈ I, n = 1, . . . , li } and find conditions
in order that they form a frame of the space H. This is an instance of the so-called
Dynamical Sampling problem, where one seeks to sample a signal at several places and
at various times, thereby exploiting time evolution, in order to reconstruct the signal
from these samples (see, e.g., [1, 2, 3, 4, 5, 12, 15]). This principle obviously deviates
from classical sampling, where a signal is to be reconstructed from its samples at a single
time (see, e.g., [19]).
As it has turned out before – and here again –, in the analysis of the Dynamical
Sampling problem one encounters connections to various different areas of mathematics,
such as, e.g., operator theory, spectral theory, harmonic analysis, and complex analysis
in the unit disk. For example (see [3]), if A is a diagonal operator in `2 (N), the system
(An f )n∈N for one vector f ∈ `2 (N) is a frame for `2 (N) if and only if the sequence of
eigenvalues of A is a so-called interpolating sequence in the unit disk and the norms of
the eigenprojections, applied to f , are uniformly bounded from above and away from
zero. In the present paper, we generalize this theorem to finite systems of vectors and
also consider non-normal operators A.
In the following, we shall introduce the reader to the motivation, the ideas, and the
details of Dynamical Sampling, describe the current state of research, and expose our
contribution in this paper.
2010 Mathematics Subject Classification. 94A20, 42C15, 30J99, 47A53.
Key words and phrases. Sampling Theory, Dynamical Sampling, Frame, Normal Operator, SemiFredholm, Strongly stable.
2
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
1.1. Motivation and idea of Dynamical Sampling. Let us assume that we are given
the task of retrieving and storing information about a signal f from a space H of functions
that vary in space in such a way that f can later be recovered from this information.
The first idea is, of course, to sample (i.e., evaluate) the function f at many convenient
places xi – hoping that the knowledge on the properties of the functions in H suffices to
recover f from the samples f (xi ). However, in real-world scenarios there are typically
many restrictions that one has to deal with. For example, the access to some of the
required places xi might be prohibited. Another problem is that sensors are usually
very expensive so that the installation of a great number of them in order to guarantee
a high-accuracy recovery becomes a crucial financial problem. The idea of Dynamical
Sampling is to avoid these obstacles by taking another dimension into account: time.
Suppose that the object that is represented by f changes over time, so that f is in
fact the evaluation at time t = 0 of a function g that depends on both space and time.
In mathematical terms, f = g(0, ·), where g(t, x) is a solution of a dynamical system. In
the simplest case, where this dynamical system is homogeneous and linear, the function
u(t) = g(t, ·), t ≥ 0, maps [0, ∞) to some function space H and satisfies u̇(t) = Bu(t),
where B is a generator of a semigroup (Tt )t≥0 of operators. The solution of this Cauchy
problem is given by u(t) = Tt u(0) = Tt f .
The idea of Dynamical Sampling now is – instead of sampling f at many places –
to sample u coarsely in space, but at various times t. This idea was for the first time
considered by Lu et al. (see [12, 15]), where the authors investigated signals obeying the
heat equation. In this case, B is the Laplace operator. In the general case – when we
sample uniformly in time –, the samples are of the form
g(nt0 , xi ) = u(nt0 )(xi ) = [Ttn0 f ](xi ),
n = 0, 1, 2, . . . , ni , i ∈ I.
If H is in fact a reproducing kernel Hilbert space with kernel K, we have
g(nt0 , xi ) = Ttn0 f, Kxi = hf, An Kxi i ,
where A := Tt∗0 . Since the original task was to recover f from the retrieved information,
the question now becomes: “Is (An Kxi )n,i complete in H?”. If one requires the recovery
to be a stable process, the question is “Is (An Kxi )n,i a frame for H?”.
1.2. Previous works on the topic. Aldroubi et al. (see [1, 2, 3, 4, 5]) considered the
problem from a somewhat more general point of view. The samples in their setting are of
the form hf, An fi i, where A is a bounded linear operator in an arbitrary Hilbert space H
and fi ∈ H. In this setting, where the objects are not fixed, the above questions become:
For which operators A, which sets I, Ni ⊂ N, and which vectors fi ∈ H, i ∈ I, is the
system (An fi )i∈I, n∈Ni complete in H or a frame for H?
In [3] the problem in the finite-dimensional setting was completely solved. For example,
if A ∈ Cm×m is diagonalizable, then (An fi )i∈I, n∈Ni is a frame for H = Cm if and only
if for each eigenprojection P of A we have that (P fi )i∈I is complete in P H. Here,
Ni = {0, . . . , ri − 1}, where ri is the degree of the A-annihilator of fi . If the operator A
is non-diagonalizable, the above statement generalizes to a somewhat more complicated
version involving the Jordan normal form of A. In the infinite-dimensional situation, the
authors considered the one-vector problem (i.e., |I| = 1) – under the additional condition
3
P
that A be diagonal (i.e., A = ∞
j=0 λj Pj , where (λj )j∈N ⊂ C is bounded and the Pj are
orthogonal projections, Pj Pk = 0 for j 6= k) with multiplicity 1 (i.e., dim Pj H = 1 for all
j ∈ N). They show (see Theorem 3.4) that (An f )n∈N is a frame for H if and only if the
sequence (λj )j∈N is uniformly separated (cf. page 9) in the
p unit disk and both sequences
−1
(aj )j∈N and (aj )j∈N are bounded, where aj := kPj f k/ 1 − |λj |2 , j ∈ N.
Recently, it was proved in [5] that the above assumption that A be diagonal is not
a restriction. In fact, this is necessary for (An fi )n∈N, i∈I to be a frame for H, at least
when A is normal and I is finite. In the case where (An f )n∈N is only required to form a
Bessel sequence, A is not necessarily diagonal. In this case, a certain measure related to
A and f has to be a so-called Carleson measure (see [14]). Instead of only considering
normal operators A (as in [3] and [2]), the authors of [5] also consider general bounded
operators A and prove that it is necessary for (An fi )n∈N, i∈I to form a frame for H that
the operator A∗ be strongly stable (i.e., (A∗ )n f → 0 as n → ∞ for each f ∈ H). If
A = Tt∗0 is derived from a semigroup as above, this means that the semigroup (Tt )t≥0 has
to be strongly stable.
1.3. Our contribution. Our contribution is two-fold. On the one hand, we generalize
the above-mentioned result in the one-vector case to the multi-vector case, that is, we
consider finite index sets I. In the one-vector case (i.e., |I| = 1), the isomorphism between
`2 (N) and the Hardy space H 2 (D) enables the use of Carleson’s theorem on interpolation
sequences to obtain the desired frame characterization. The attempt of proving a similar
result for |I| > 1 with the same methods fails terribly because Carleson’s theorem cannot
be used directly. As a consequence, the problem becomes much more difficult and new
techniques have to be found. Our strategy consists of a careful analysis of the Dynamical
Sampling problem for normal operators in Section 3, thereby deriving some technical
lemmas. They will be the key to prove our main theorem in this section (cf. Theorem
3.8). This theorem characterizes those normal operators A and vectors fi (the index set
I being finite) for which (An fi )n∈N, i∈I is a frame for H. As it turns out, the condition
that the eigenvalue sequence Λ = (λj )j∈N be uniformly separated (as in the one-vector
case) has to be replaced by the weaker condition that Λ be a finite union of uniformly
separated sequences. From here, the question naturally arises, how many uniformly
separated sequences are contained in Λ. An obvious first conjecture is that the answer is
|I|, which would fit to the one-vector case. However, this is a non-trivial question and we
leverage another characterization result, Theorem 3.3, for showing in Theorem 3.12 that
the conjecture indeed is true: the minimal number of uniformly separated subsequences
in a partition of Λ (which we call the index of Λ) is bounded from above by |I| – just as
the dimensions dim Pj H. This upper bound on the index is sharp (see Remark 3.13).
On the other hand, we prove new results for the case where A is not assumed to be
normal. In fact, if A∗ = Tt0 is an instance of a semigroup (Tt )t≥0 as in the above motivation for the Dynamical Sampling setting, the operator A is bounded but not normal,
in general. It is clear that in this more general case, we loose the strong instrument of
the highly-developed spectral theory for normal operators, so that the problem becomes
much more involved. However, we show that also for non-normal operators A and finite
index sets it is necessary for (An fi )n∈N, i∈I to be a frame for H that the spectrum of A
4
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
be contained in the closed unit disk and that its intersection with the unit circle is nonempty. We furthermore prove a characterization result (see Proposition 4.3) in which we
do not assume that the index set is finite. From this characterization it follows that it is
necessary for (An fi )n∈N, i∈I to be a Bessel sequence that
ASA∗ = S − F,
where S and F are the frame operators of (An fi )n∈N, i∈I and (fi )i∈I , respectively. This
relation lays the foundations for Theorem 4.6 which states in particular that for finite
index sets I (which we shall assume from here) the eigenvalues of A∗ in D either fill
the whole open unit disk D or are discrete in D with finite multiplicities. Although the
first case seems to be a rarity, we provide an example in which this case occurs and
(An fi )n∈N, i∈I is a frame for H (cf. Example 4.11). The system in this example is in fact
an orthonormal basis and we prove that this holds in general: in order that (An fi )n∈N, i∈I
be a Riesz basis of H it is necessary that each point in the open unit disk is an eigenvalue
of A∗ . Using this, it is shown that (An fi )n∈N, i∈I can never be a Riesz basis of H when
A∗ = Tt0 is the instance of a semigroup (see Corollary 4.14).
1.4. Outline. The present paper is organized as follows. In Section 2 we provide the
necessary background from frame theory, operator theory, spectral theory, and the theory
of sequences in the open unit disk. Section 3 contains our main results concerning
Dynamical Sampling on finite index sets with normal operators, including the abovementioned characterization and the estimate on the index of the eigenvalue sequence. In
Section 4, we drop the requirement that A be normal and provide our results, summarized
above, for this more general setting. They show that there are some similarities to the
normal situation, but also vast differences.
2. Preliminaries
First of all, let us fix the notation that we shall use throughout this paper. By N we
denote the set of the natural numbers including zero. Unit circle and open unit disk in
C are denoted by T and D, respectively, i.e.,
T = {z ∈ C : |z| = 1}
and
D = {z ∈ C : |z| < 1}.
The open ball in C with radius r > 0 and center z ∈ C will be denoted by Br (z). The p-th
Hardy space on the unit disk, 1 ≤ p ≤ ∞, is denoted by H p (D). Recall
P that especially
n
H 2 (D) consists of those functions that have a representation ϕ(z) = ∞
n=0 cn z , z ∈ D,
where c = (cn )n∈N ∈ `2 (N), and that kϕkH 2 (D) = kck2 .
Throughout, H stands for a separable Hilbert space. By L(H) we denote the set
of all bounded operators from H to itself which are defined on all of H. The kernel
(i.e., the null-space) and the range (i.e., the image) of T ∈ L(H) are denoted by ker T
and ran T , respectively. The spectrum of T ∈ L(H) (i.e., the set of all λ ∈ C for which
T −λ = T −λ Id is not boundedly invertible) is denoted by σ(T ), the set of all eigenvalues
of T (usually called the point spectrum of T ) by σp (T ). The continuous spectrum of T
is the set of all λ ∈ σ(T ) \ σp (T ) for which ran(T − λ) is dense in H. It is denoted by
σc (T ).
5
2.1. Vector systems in Hilbert spaces. Let F = (fn )n∈N be a sequence of vectors in
the Hilbert space H. The analysis operator of F is then defined as T : H ⊃ D(T ) → `2 (N),
T f := (hf, fn i)n∈N ,
f ∈ D(T ),
where the domain D(T ) of T is given by
D(T ) := f ∈ H : (hf, fn i)n∈N ∈ `2 (N) .
It is well known (and straightforward to prove) that T is always a closed operator (i.e.,
its graph is closed in H ⊕ `2 (N)). Hence, by the closed graph theorem, we have T ∈ L(H)
if and only if D(T ) = H. In this case, the sequence F is called a Bessel sequence. That
is, F is a Bessel sequence if and only if there exists some B > 0 (a so-called Bessel bound
of F) such that
∞
X
|hf, fn i|2 ≤ Bkf k2 ,
f ∈ H.
n=0
Let F = (fn )n∈N ⊂ H be a Bessel sequence. The adjoint T ∗ of T is then called the
synthesis operator of F. It can be represented by
∞
X
T ∗c =
cn fn ,
c = (cn )n∈N ∈ `2 (N).
n=0
The operator S := T ∗ T is called the frame operator of F. We have
∞
X
Sf =
hf, fn ifn ,
f ∈ H.
n=0
It is evident that S is a bounded non-negative selfadjoint operator in H. The Bessel
sequence F is said to be a frame for H if its frame operator S is boundedly invertible,
i.e., there exist A, B > 0 such that Akf k2 ≤ hSf, f i ≤ Bkf k2 for all f ∈ H. Equivalently,
∞
X
Akf k2 ≤
|hf, fn i|2 ≤ Bkf k2 ,
f ∈ H.
(2.1)
n=0
More generally, if (2.1) holds for all f ∈ HF := span{fn : n ∈ N}, then F is called a
frame sequence. The bounds A and B are called (lower and upper, respectively) frame
bounds of F. If F is a frame for H, the frame relation (2.1) immediately yields that
(fn )n∈N is complete in H, i.e., HF = H.
Another object associated with a vector sequence F = (fn )n∈N is its Gram matrix
G = (hfn , fm i)n,m∈N which is obviously hermitian. A Bessel sequence F = (fn )n∈N ⊂ H
is called a Riesz sequence if there exist A, B > 0 such that Akck22 ≤ kT ∗ ck2 ≤ Bkck22 for
each c ∈ `2 (N). A Riesz basis of H is a complete Riesz sequence.
In this paper we shall make use of the following collection of basic statements. Proofs
of them can be found in [6].
Theorem 2.1. Let F = (fn )n∈N be a sequence of vectors in H. Then the following
statements hold.
(i) F is a Bessel sequence if and only if G defines an operator in L(`2 (N)).
(ii) F is a frame sequence if and only if G defines an operator in L(`2 (N)) with closed
range.
6
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
(iii) The Bessel sequence F is a frame for H if and only if T ∗ is surjective.
(iv) The Bessel sequence F is a Riesz sequence if and only if T is surjective.
2.2. Normal and diagonal operators. Recall that an operator A ∈ L(H) is said to
be normal if A∗ A = AA∗ . It is well known (see, e.g., [7]) that a normal operator induces
a spectral measure E which maps Borel sets in C to orthogonal projections on H, thereby
satisfying the following condition:
X
E(∆)f =
E(∆j )f,
f ∈ H,
j∈J
S
where J is at most countable, the ∆j are mutually disjoint, and ∆ = j∈J ∆j . A spectral
measure allows integration of measurable
functions φ on σ(A). The result is a (possibly
R
is given by D(φ(A)) = {f ∈ H :
unbounded) normal operator φ(A) = φ dE. Its domain
R
φ ∈ L2 (dµf )}, where µf = kE(·)f k2 . We have A = z dE and (cf. [7, Theorem X.4.7])
Z
2
kφ(A)f k = |φ|2 dµf , f ∈ D(φ(A)).
(2.2)
In the sequel, we will often be dealing with diagonal operators – a special class of
normal operators – which we define as follows.
P
Definition 2.2. A diagonal operator in H is of the form A =
j∈J λj Pj (the series
converging in the strong operator topology), where J is a finite or countable index set,
(λj )j∈J ⊂ C a bounded sequence of scalars, and
P (Pj )j∈J a sequence of orthogonal projections with Pj Pk = 0 for j 6= k. The series j∈J λj Pj is called a normal form of A if
P
λj 6= λk for j 6= k and j∈J Pj = Id. The multiplicity of a diagonal operator A is defined
by
mult(A) := max{dim Pj H : j ∈ J},
where (Pj )j∈J is the sequence of orthogonal projections in a normal form of A. If the
maximum should not exist, we set mult(A) := ∞ and say that A has infinite multiplicity.
The normal form of a diagonal operator is obviously unique up to permutations of J.
Moreover, it is clear that the λj in the normal form of A are the distinct eigenvalues of A
and that Pj projects onto the eigenspace ker(A − λj ). Note that every diagonal operator
is bounded and normal.
Lemma 2.3. A normal operator A ∈ L(H) is a diagonal operator if and only if
E(σ(A) \ σp (A)) = 0.
P
Proof. Let A = j∈J λj Pj be a diagonal operator in normal form. Then it is clear that
P
σp (A) = {λj : j ∈ J}. Moreover, the spectral measure of A is given by E = j∈J δλj Pj
R
P
P
because z d( j∈J δλj Pj ) =
j∈J λj Pj = A. Thus, E(σ(A) \ σp (A)) = 0. For the
converse, we note that since H is separable and the operator A is normal, the set of its
distinct eigenvalues is countable. Thus, σp (A) = {λj : j ∈ J}, where J ⊂ N and λj 6= λk
for j 6= k. If E(σ(A) \ σp (A)) = 0, then for f ∈ H we have
Z
Z
XZ
X
Af =
z dEf =
z dEf =
z dEf =
λj E({λj })f,
σ(A)
σp (A)
j∈J
{λj }
j∈J
7
which shows that A is a diagonal operator.
P
As it was seen in the proof of Lemma 2.3, if A = j∈J λj Pj is the normal form of the
diagonal operator A, the spectral measure E of A is given by
X
E=
δλj Pj .
j∈J
Hence, for any vector h ∈ H we have
µh =
X
kPj hk2 δλj .
(2.3)
j∈J
2.3. Semi-Fredholm operators. An operator T ∈ L(H) is said to be upper semiFredholm, if ker T is finite-dimensional and ran T is closed. The operator T is called
lower semi-Fredholm, if codim ran T < ∞ (in this case, the range of T is automatically
closed). T is called semi-Fredholm if it is upper or lower semi-Fredholm and Fredholm
if it is both upper and lower semi-Fredholm. In all cases, one defines the nullity and
deficiency of T by
nul(T ) := dim ker T
and
def(T ) := codim ran T.
The index of T is defined by
ind(T ) := nul(T ) − def(T ).
This value might be a positive or negative integer or ±∞. It is, moreover, easily seen that
T is upper semi-Fredholm if and only if T ∗ is lower semi-Fredholm. We have def(T ∗ ) =
nul(T ) and nul(T ∗ ) = def(T ) and thus ind(T ∗ ) = − ind(T ). The next theorem shows
that the semi-Fredholm property of operators is stable under compact perturbations.
Theorem 2.4 ([11, Theorem IV.5.26]). If K ∈ L(H) is compact and T ∈ L(H) is upper
(lower ) semi-Fredholm, then T + K is upper (lower, respectively ) semi-Fredholm with
ind(T + K) = ind(T ).
For the following lemma see [13, Theorems III.16.5, III.16.6, and III.16.12].
Lemma 2.5. Let S, T ∈ L(H). Then the following statements hold.
(i) If ST is upper semi-Fredholm, then so is T .
(ii) If S and T are upper semi-Fredholm, then so is ST and
ind(ST ) = ind(S) + ind(T ).
(2.4)
While Theorem 2.4 deals with compact perturbations, the next theorem (also known
as the (extended) punctured neighborhood theorem (see [10, Theorems 4.1 and 4.2])) is
concerned with perturbations of the type λ Id, where |λ| is small.
Theorem 2.6. Let T ∈ L(H) be upper semi-Fredholm and put
k := dim [ker T /(ker T ∩ R∞ (T ))] ,
T∞
where R∞ (T ) := n=0 ran(T n ). Then there exists ε > 0 such that for 0 < |λ| < ε the
following statements hold.
(i) T − λ is upper semi-Fredholm.
8
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
(ii) ind(T − λ) = ind(T ).
(iii) nul(T − λ) = nul(T ) − k.
(iv) def(T − λ) = def(T ) − k.
2.4. Sequences in the unit disk and the pseudo-hyperbolic metric. A sequence
Λ = (λj )j∈N in the open unit
D is called interpolating for H p (D), 1 ≤ p < ∞, if for
Pdisk
∞
each sequence (dj )j∈N with j=0 |dj |p (1 − |λj |2 ) < ∞ there exists some ϕ ∈ H p (D) such
that ϕ(λj ) = dj for all j ∈ N. Λ is called interpolating for H ∞ (D), if for each sequence
(dj )j∈N ∈ `∞ there exists some ϕ ∈ H ∞ (D) such that ϕ(λj ) = dj for all j ∈ N. We define
the linear evaluation operator
TΛ : H 2 (D) ⊃ D(TΛ ) → `2 (N),
TΛ ϕ := (εj ϕ(λj ))j∈N ,
(2.5)
p
where εj := 1 − |λj |2 and
D(TΛ ) = ϕ ∈ H 2 (D) : (εj ϕ(λj ))j∈N ∈ `2 (N) .
Note that εj ϕ(λj ) = hϕ, Kλj iH 2 (D) , where
∞
X
p
n
2
Kλ (z) = 1 − |λ|
λ zn =
n=0
p
1 − |λ|2
,
1 − λz
λ, z ∈ D,
(2.6)
is the normalized reproducing kernel of H 2 (D). Hence, the operator TΛ is the analysis
operator corresponding to the sequence (Kλj )j∈N . And as every analysis operator is
closed on its natural domain, it follows from the closed graph theorem that TΛ is a
bounded operator from H 2 (D) to `2 (N) if and only if D(TΛ ) = H 2 (D).
Lemma 2.7. If 1 ∈ D(TΛ ), then idn ∈ D(TΛ ) for all n ∈ N. In particular, TΛ is densely
defined. If, in addition, λi 6= λj for i 6= j, then ran TΛ is dense in `2 (N).
Proof. First, 1 ∈ D(TΛ )P
means that Λ is aPBlaschke sequence, i.e., (εj )j∈N ∈ `2 (N). Thus,
∞
n
2
2
2n ≤
for n ∈ N we have that ∞
j=0 εj < ∞. That is, id ∈ D(TΛ ). If λi 6= λj
j=0 εj |λj |
for i 6= j, for fixed i ∈ N let Bi be the Blaschke product of (λj )j6=i , i.e.,
Y λj − z |λj |
Bi (z) = z k
,
1 − λj z λj
j6=i
where k ∈ {0, 1} and k = 0 iff λj 6= 0 for all j 6= i. Set fi := (εi Bi (λi ))−1 Bi , i ∈ N. Then
fi ∈ H ∞ (D) ⊂ H 2 (D) (see, e.g., [16, Theorem 15.21]). Moreover, fi ∈ D(TΛ ) and TΛ fi is
the i-th standard basis vector of `2 (N). Hence, ran TΛ is dense in `2 (N).
Recall that the pseudo-hyperbolic metric % on the open unit disk is defined by
z−w , z, w ∈ D.
%(z, w) := 1 − zw Since
|1 − zw|2 = |z − w|2 + (1 − |z|2 )(1 − |w|2 ),
we always have %(z, w) < 1. It is well known that % is indeed a metric on D.
(2.7)
9
Lemma 2.8. For z ∈ D put sz =
holds:
p
1 − |z|2 . Then for z, w ∈ D the following relation
kKz − Kw k2H 2 (D) = (2 − sz sw )%(z, w)2 − 1 − %(z, w)2
In particular,
kKz − Kw kH 2 (D) ≤
√
(sz − sw )2
.
sz sw
2 %(z, w).
Proof. Using 1 − %(z, w)2 = s2z s2w /|1 − zw|2 (see (2.7)), we see that
sz sw
sz sw
kKz − Kw k2H 2 (D) = 2 − 2 RehKz , Kw i = 2 − 2 Re
(1 − Re(zw))
=2− 2
1 − zw
|1 − zw|2
sz sw
=2−
(2 − 2 Re(zw) − 2sz sw ) − 2 1 − %(z, w)2
2
|1 − zw|
sz sw
= 2%(z, w)2 −
2 + |z − w|2 − |z|2 − |w|2 − 2sz sw
2
|1 − zw|
sz sw
2
(sz − sw )2 + |z − w|2 ,
= 2%(z, w) −
2
|1 − zw|
which proves the claim.
A sequence Λ = (λj )j∈N in the open unit disk is called separated if
inf %(λj , λk ) > 0.
j6=k
The sequence Λ is called uniformly separated if
Y
inf
%(λj , λk ) > 0.
n∈N
j6=k
The following theorem is due to Shapiro and Shields [17].
Theorem 2.9. Let 1 ≤ p ≤ ∞. For a sequence Λ = (λk )k∈N ⊂ D the following statements
are equivalent.
(i) Λ is an interpolating sequence for H p (D).
(ii) The evaluation operator TΛ is defined on H 2 (D) and is onto.
(iii) The sequence (Kλj )j∈N is a Riesz sequence in H 2 (D).
(iv) Λ is uniformly separated.
Note that the equivalence of (ii) and (iii) in Theorem 2.9 simply follows from the fact
that TΛ is the analysis operator of the sequence (Kλj )j∈N . The following two theorems
can be found in [9].
Theorem 2.10. Let Λ = (λj )j∈N ⊂ D. Then the following conditions are equivalent.
(i) Λ is a finite union of uniformly separated sequences.
(ii) D(TΛ ) = H 2 (D).
Theorem 2.11. Let Λ = (λj )j∈N ⊂ D. Then the following statements are equivalent.
(i) Λ is uniformly separated.
(ii) Λ is separated and D(TΛ ) = H 2 (D).
10
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
Note that Theorem 2.11 is not formulated as a theorem in [9], but is hidden in the
proof of the implication (iii)⇒(i) of the main theorem. It immediately implies the next
corollary.
Corollary 2.12. Let (λj )j∈N be a sequence in D such that λj 6= λk for j 6= k. If (λj )j≥n
is uniformly separated, then also (λj )j∈N is uniformly separated.
Definition 2.13. Let Λ = (λj )j∈N be a finite union of separated sequences. The index
of Λ is then defined by
)
(
n
[
ind(Λ) = min n ∈ N : N =
Jk , (λj )j∈Jk is separated for each k = 1, . . . , n .
k=1
For z ∈ D and r > 0 by Br (z) we denote the pseudo-hyperbolic ball (%-ball) in D of
radius r and center z, i.e.,
Br (z) = {λ ∈ D : %(λ, z) < r}.
Note that Br (z) = Br
0
0 (z )
with r0 < r and z 0 = tz, where t < 1.
Lemma 2.14. Let Λ = (λj )j∈N be a sequence in D. For r ∈ (0, 1) and z ∈ D define
J(r, z) := {j ∈ N : λj ∈ Br (z)}
and
mr := sup |J(r, z)| ,
z∈D
and assume mr0 < ∞ for some r0 ∈ (0, 1). Then Λ is a finite union of mr0 separated
sequences (or less ) and
ind(Λ) = inf{mr : r ∈ (0, r0 ]}.
(2.8)
Proof. For J ⊂ N define ΛJ := (λj )j∈J and set
S M := mr0 . We will define subsets
J1 , . . . , Jm , m ≤ M , recursively such that N = m
k=1 Jk and each ΛJk is separated. For
the definition of J1 , set j0 := 0 and once j0 , . . . , js are chosen, we pick
(
)
s
[
js+1 := min j > js : λj ∈
/
Br (λji ) .
i=1
Note that js+1 is well defined due to the assumption that M < ∞. In other words, λjs+1
is the first element of the sequence Λ that does not belong to any of the balls of radius
r around the previously chosen elements. Put J1 := {js : s ∈ N}. It is clear that ΛJ1 is
separated (by r) and that all the λj that have not been chosen due to this process belong
to some Br (λjs ).
If N \ J1 is finite, we are finished. If not, proceed as before with N \ J1 instead of N
to find an infinite set J2 ⊂ N \ J1 such that ΛJ2 is separated (by r). Continuing in this
way, the process either terminates after m < M − 1 steps (in which case we are done) or
we obtain M − 1 separated sequences ΛJ1 , . . . , ΛJM −1 . In this case, put
JM := N \
M
−1
[
Jk .
k=1
Let us prove that ΛJM is separated. For this, let j ∈ JM be arbitrary. Then, as a result
S −1
of the construction process, λj ∈ M
k=1 Br (λik ), where ik ∈ Jk , k = 1, . . . , M − 1. Thus,
11
we have that j, i1 , . . . , iM −1 ∈ J(r, λj ) such that J(r, λj ) = {j, i1 , . . . , iM −1 }. Therefore,
%(λj , λl ) ≥ r for all l ∈ JM , l 6= j. Hence, ΛJM is indeed separated.
It remains to prove the relation (2.8). For this, put m0 := inf{mr : r ∈ (0, r0 ]}. It
and
is clear that m0 = mr1 for some r1 ≤ r0 (note that r 7→ mr is non-decreasing)
S
that, therefore, n := ind(Λ) ≤ m0 . There exist J1 , . . . , Jn ⊂ N with N = nk=1 Jk such
that ΛJk is separated for each k = 1, . . . , n. Without loss of generality, let each ΛJk
be separated by r1 . From m0 = mr1 /2 we conclude that there exists some z ∈ D such
that |J(r1 /2, z)| = m0 . Suppose that n < m0 . Then there exists some Jk that contains
at least two of the m0 elements of J(r1 /2, z), say, j1 and j2 . But then %(λj1 , λj2 ) ≤
%(λj1 , z) + %(z, λj2 ) < r1 , contradicting the fact that ΛJk is separated by r1 .
3. Dynamical Sampling with normal operators
We will now start investigating the sequences of the form (An fi )n∈N, i∈I where A is
a bounded normal operator in H, I an at most countable index set, and (fi )i∈I ⊂ H.
Throughout, we set
A := A(A, (fi )i∈I ) := (An fi )n∈N, i∈I .
The spectral measure of A will be denoted by E.
The fact that condition (i) in the next theorem is necessary for A to be a frame for
H has already been proved in [5]. Since in [5] it was not assumed that A is normal,
this result also plays a certain role in the next section, where we consider non-normal
operators in the Dynamical Sampling setting.
Theorem 3.1. The following two conditions are equivalent to each other and necessary
for A to be a frame for H.
(i) (A∗ )n f → 0 as n → ∞ for any f ∈ H.
(ii) σ(A) ⊂ D and E(T) = 0.
Proof. Assume that A is a frame for H with lower frame bound α > 0. Then for f ∈ H
and any m ∈ N we have
∗ m
2
αk(A ) f k ≤
∞ X
X
n=0 i∈I
n+m
|hf, A
2
fi i| =
∞ X
X
|hf, An fi i|2 .
(3.1)
n=m i∈I
The latter series converges for m = 0 since A is a Bessel sequence. Hence, (A∗ )m f → 0
as m → ∞. Note that this is equivalent to An f → 0 as n → ∞ since A is normal.
(i)⇒(ii). If (A∗ )n f → 0 as n → ∞ for each f ∈ H, let f ∈ E(C \ D)H. If we define
the measure µf := kE(·)f k2 and use (2.2), we obtain
Z
n
2
kA f k =
|z|2n dµf ≥ µf (C \ D) = kf k2 .
C\D
Hence, E(C \ D) = 0 which is clearly equivalent to (ii).
R
(ii)⇒(i). Let E(C\D) = 0. Then, for any f ∈ H we have kAn f k2 = D |z|2n dµf . Since
1 ∈ L1 (µf ), we conclude from Lebesgue’s dominated convergence theorem that An f → 0
as n → ∞.
12
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
The next lemma is a variant of a statement in [2]. The “in particular”-part was proved
in [5]. For r > 0 we define Dr = {z ∈ C : |z| < r}.
Lemma
P 3.2. Assume2 that A is a frame for H with lower frame bound α > 0. If
Mr := i∈I kE(Dr )fi k < ∞ for some r ∈ (0, 1) and m ∈ N is the smallest integer such
that r2m Mr < α(1 − r2 ), then
E(Dr )H = span{An E(Dr )fi : 0 ≤ n ≤ m − 1, i ∈ I}.
In particular, if I is a finite set, then for any r ∈ (0, 1) we have that Mr < ∞ and the
space E(Dr )H is finite-dimensional.
Proof. Put fi (r) := E(Dr )fi , i ∈ I. Let m ∈ N be as in the lemma and let g ∈ E(Dr )H
such that hg, An fi (r)i = 0 for all 0 ≤ n ≤ m − 1 and all i ∈ I. Then from
αkgk2 ≤
∞
XX
|hg, An fi (r)i|2 =
≤ kgk
∞
XX
n
2
∞
XX
2
kA fi (r)k = kgk
∞ Z
XX
|z|2n dµfi (z)
i∈I n=m Dr
i∈I n=m
≤ kgk2
|hg, An fi (r)i|2
i∈I n=m
i∈I n=0
2
∞
XX
r2n kfi (r)k2 =
i∈I n=m
r2m
Mr kgk2
1 − r2
and the choice of m we obtain g = 0.
For the rest of this section we shall restrict ourselves to diagonal operators A. The
reason for this is that the operator A is necessarily diagonal if A is a frame for H and I
is finite (see Theorem 3.8).
L
For an at most countable index set I we define HI2 := i∈I H 2 (D). This is the space
P
of tuples φ = (ϕi )i∈I , where ϕi ∈ H 2 (D) for each i ∈ I, such that i∈I kϕi k2H 2 (D) < ∞.
P
One defines kφkH 2 := ( i∈I kϕi k2H 2 (D) )1/2 .
I
In addition, we shall write the tensor product of a sequence ~y = (yi )i∈I ⊂ C and a
function ϕ ∈ H 2 (D) as ~y ϕ (i.e., (~y ϕ)(z) = (yi ϕ(z))i∈I , z ∈ D). The result is an element
of HI2 if and only if ~y ∈ `2 (I). In this case,
k~y ϕkH 2 = k~y k2 kϕkH 2 (D) .
I
For a sequence Λ = (λj )j∈N ⊂ D in the open unit disk we agree to write
q
εj := 1 − |λj |2 ,
j ∈ N.
(3.2)
(3.3)
Although εj depends on Λ, it will always be clear which Λ it refers to.
P
Theorem 3.3. Let A = ∞
j=0 λj Pj be a diagonal operator in normal form with (λj )j∈N ⊂
D and let fi ∈ H, i ∈ I. For j ∈ N we put nj := dim Pj H (where possibly nj = ∞) and
nj
Lj := {1, . . . , nj }. Moreover, let (ejl )l=1
be an orthonormal basis of Pj H and for l ∈ Lj
−1
define ~yjl := εj (hejl , fi i)i∈I as well as φjl := ~yjl Kλj . Then the following statements
hold.
13
(i) A is a Bessel sequence in H if and only if ~yjl ∈ `2 (I) (i.e., φjl ∈ HI2 ) for each
j ∈ N and each l ∈ Lj and Φ = (φjl )j∈N, l∈Lj is a Bessel sequence in HI2 . In this
case, the Bessel bounds of A and Φ coincide.
(ii) A is a frame for H if and only if ~yjl ∈ `2 (I) for each j ∈ N and each l ∈ Lj and
Φ = (φjl )j∈N, l∈Lj is a Riesz sequence in HI2 . In this case, the frame bounds of A
coincide with the Riesz bounds of Φ.
Proof. In the following we will often makePuse of the unitary operator U : `2 (I ×N) → HI2 ,
defined by U x = (ϕi )i∈I , where ϕi (z) = n∈N xin z n , z ∈ D.
Assume that A is a Bessel sequence in H with Bessel bound β > 0. Then for h ∈ H
we have that
∞ X
X
|hh, An fi i|2 ≤ βkhk22 .
n=0 i∈I
Let j ∈ N and l ∈ Lj . Since for h = ejl we have
∞ X
X
n
2
|hh, A fi i| =
n=0 i∈I
∞ X
X
|λj |2n |hejl , fi i|2 =
n=0 i∈I
X |hejl , fi i|2
i∈I
1 − |λj |2
,
P
2
yjl ∈ `2 (I). Let ψ = (ψi )i∈I ∈ HI2 and put
it follows that i∈I ε−2
j |hejl , fi i| ≤ β, i.e., ~
−1
2
x := U ψ ∈ ` (I × N). Denote the synthesis operator of A by T . Then for each j ∈ N
and l ∈ Lj we have
+
*
∞
∞
XX
XX
X
n
xin A fi , ejl =
xin λnj hfi , ejl i =
ψi (λj )hfi , ejl i
hT x, ejl i =
i∈I n=0
=
X hfi , ejl i i∈I
εj
i∈I n=0
ψi , Kλj =
X
i∈I
ψi ,
i∈I
hejl , fi i
Kλj
εj
= hψ, φjl i.
P∞ Pnj
Pnj
P
2
2
2
Thus, ∞
j=0
j=0
l=1 |hT x, ejl i| = kT xk , which implies that the
l=1 |hψ, φjl i| =
sequence (φjl )j∈N, l∈Lj is a Bessel sequence in HI2 . Let C denote its analysis operator.
Then the above relation shows that T = CU . In particular, the Bessel bounds of both
sequences coincide. Moreover, if A is a frame for H, then C = T U ∗ is onto, meaning that
(φjl )j∈N, l∈Lj is indeed a Riesz sequence and that its lower Riesz bound coincides with
the lower frame bound of (An fi )n∈N, i∈I .
Assume conversely that ~yjl ∈ `2 (I) for each j ∈ N and l ∈ Lj and that (φjl )j∈N, l∈Lj
is a Bessel sequence in HI2 with Bessel bound β > 0. If x ∈ `2 (I × N) with only
finitely many non-zero entries and ψ = U x, then hT x, ejl i = hψ, φjl i, that is, kT xk2 =
P∞ Pnj
2
2
2
j=0
l=1 |hψ, φjl i| ≤ βkψk = βkxk . Thus, A is a Bessel sequence. If, in addition,
(φjl )j∈N, l∈Lj is a Riesz sequence, then T = CU is onto, which means that A is a frame
for H.
3.1. Singleton index sets. If the index set I only contains one element, the system A
has the form (An f )n∈N , where f ∈ H. The next theorem has been proved in [3, Theorem
3.14]. We would like to mention that the operator in [3, Theorem 3.14] was actually
14
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
assumed to be selfadjoint. However, the proof goes through in the same way with a
normal operator.
P
Theorem 3.4. Let A = ∞
j=0 λj Pj be a diagonal operator in normal form and f ∈ H.
n
Then (A f )n∈N is a frame for H if and only if the following statements hold:
(i) (λj )j∈N is a uniformly separated sequence in D.
(ii) dim Pj H = 1 for all j ∈ N.
(iii) There exist α, β > 0 such that
α ≤
kPj f k2
≤ β
1 − |λj |2
for all j ∈ N.
When comparing the sequences (An f )n∈N and ((A∗ )n f )n∈N it might be surprising at
first glance that the following proposition holds.
Proposition 3.5. For f ∈ H the following statements hold.
(i) (An f )n∈N is a Bessel sequence if and only if ((A∗ )n f )n∈N is a Bessel sequence.
(ii) (An f )n∈N is a frame sequence if and only if ((A∗ )n f )n∈N is a frame sequence.
(iii) (An f )n∈N is a frame for H if and only if ((A∗ )n f )n∈N is a frame for H.
Proof. Let GA and GA∗ denote the Gram matrices of (An f )n∈N and ((A∗ )n f )n∈N , respectively. Moreover, denote the (n, m)-th entry of GA (GA∗ ) by anm (A) (anm (A∗ ),
respectively). Then, using the normality of A gives
anm (A∗ ) = h(A∗ )n f, (A∗ )m f i = hAm f, An f i = anm (A).
From this, it almost immediately follows that GA defines a bounded operator if and only
if GA∗ does and that ran GA is closed if and only if ran GA∗ is closed. Thus, (i) and (ii)
are proved (cf. Theorem 2.1). The claim (iii) follows from Theorem 3.4 if A is a diagonal
operator. However, the necessity of A being a diagonal operator for (An f )n∈N to be a
frame is a consequence of [5] (see also Theorem 3.8).
In the next lemma we formulate the characterization theorems from [14] for the case
of a diagonal operator.
P
Lemma 3.6. Let A = ∞
j=0 λj Pj be a diagonal operator in normal form and f ∈ H.
Put J := {j ∈ N : Pj f 6= 0}. Then the following statements are equivalent.
(i) The sequence (An f )n∈N is a Bessel sequence in H.
(ii) λj ∈ D for all j ∈ J and there exists a constant C > 0 such that
2
∞ X
X
2
n
2
c
λ
kP
f
k
for all c ∈ `2w (J),
j
j
j
≤ Ckck`2w (J)
n=0 j∈J
where `2w (J) is the weighted `2 -space over J with w = (kPj f k2 )j∈J .
(iii) λj ∈ D for all j ∈ J and there exists a constant C > 0 such that
X
|ϕ(λj )|2 kPj f k2 ≤ Ckϕk22
for all ϕ ∈ H 2 (D).
j∈J
15
(iv) λj ∈ D for all j ∈ J and there exists a constant C > 0 such that for all r > 0 and
all z ∈ T we have
X
kPj f k2 ≤ Cr.
j∈J : λj ∈Br (z)
(v) λj ∈ D for all j ∈ J and there exists a constant C > 0 such that
X kPj f k2
C
≤
for all z ∈ D.
|1 − zλj |2
1 − |z|2
j∈J
P
Proof. Since A is a diagonal operator, we have that µf = j∈N δλj kPj f k2 (see (2.3)).
Assume that (i) holds. Then supp(µf ) ⊂ D and µf |T is absolutely continuous with
respect to the arc length measure on T (see [14]). Therefore, we have Pj f = 0 if λj ∈
/ D.
But this also holds for λj ∈ T due to the absolute continuity of µf |T. Summarizing, we
have λj ∈ D for all j ∈ J. Since the latter is necessary for all conditions (i)–(v), we shall
assume it for the rest of this proof.
We will use the various characterizations for (An f )n∈N to be a Bessel sequence from
[14], where it is proved that (i) is equivalent to each of the following conditions:
(ii’) The sequence (idn )n∈N is a Bessel sequence in L2 (µf ).
(iii’) H 2 (D) ⊂ L2 (µf ) and this embedding is continuous.
(iv’) µf |D is a Carleson measure, that is, there exists a constant C > 0 such that for
all r > 0 and z ∈ T we have µf (Br (z)) ≤ Cr.
R 1−|z|2
(v’) supz∈D D |1−zλ|
2 dµf (λ) < ∞.
In what follows, we shall show that in the given situation (x’) is equivalent to (x), where
x ∈ {ii, iii, iv, v}. In fact, (iv’)⇔(iv) is obvious and the equivalences (iii’)⇔(iii) and
(v’)⇔(v) immediately follow from
Z
X
|ϕ|2 dµf =
|ϕ(λj )|2 kPj f k2 ,
ϕ ∈ L2 (µf ).
(3.4)
j∈J
It remains to prove (ii’)⇔(ii). As a consequence of (3.4),the mapping Φ : L2 (µf ) → `2w (J),
defined by
Φϕ := (ϕ(λj ))j∈J , ϕ ∈ L2 (µf ),
is an isometric isomorphism. Now, condition (ii’)
only if there exists some
P holds if and
n 2
C > 0 such that for every ϕ P
∈ L2 (µf ) we have ∞
|hϕ,
id
i|
≤ Ckϕk22 . Equivalently,
n=0
n 2
2
for every c ∈ `2w (J) we have ∞
n=0 |hc, Φ id i| ≤ Ckck`2 (J) . But this is (ii).
w
Remark 3.7. It follows from [14] that the constant C in (ii) and (iii) can be chosen to
be equal to the Bessel bound of (An f )n∈N .
3.2. Finite index sets. In this subsection we present our main results concerning Dynamical Sampling with normal operators on finite index sets. We will start right off with
the main theorem. To formulate it, for a sequence Λ in D we define the operator
M
TΛ,I :=
TΛ ,
i∈I
16
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
where TΛ is the evaluation operator from (2.5). That is, TΛ,I is a linear operator mapping
from D(TΛ,I ) = D(TΛ )|I| ⊂ HI2 (here, HI2 = (H 2 (D))|I| ) to (`2 (N))|I| = `2 (I × N). For a
Bessel sequence E in H we let CE denote the analysis operator of E.
Theorem 3.8. If |I| is finite, then the system A = (An fi )n∈N, i∈I is a frame for H if
and only if the following conditions are satisfied:
P
(i) A = ∞
j=0 λj Pj is a diagonal operator (in normal form ) with mult(A) ≤ |I|.
(ii) Λ = (λj )j∈N is a finite union of uniformly separated sequences in D.
(iii) There exist α, β > 0 such that for all j ∈ N and all h ∈ Pj H we have
X
α(1 − |λj |2 )khk2 ≤
|hh, Pj fi i|2 ≤ β(1 − |λj |2 )khk2 .
(3.5)
i∈I
(iv) ran TΛ,I + ker CE∗ = `2 (I × N), where E := ((1 − |λj |2 )−1/2 Pj fi )j∈N, i∈I .
Remark 3.9. (a) The necessity of condition (i) for A to be a frame for H was already
proved in [5].
(b) Condition (iii) in Theorem 3.8 means that for each j ∈ N the finite system ((1 −
|λj |2 )−1/2 Pj fi )i∈I is a frame for Pj H with frame bounds α and β. Since the frame
bounds are independent of j ∈ N, condition (iii) is equivalent to saying that the system
E = ((1 − |λj |2 )−1/2 Pj fi )j∈N, i∈I is a frame for H with frame bounds α and β.
(c) Since E is a frame for H, the relation ran TΛ,I + ker CE∗ = `2 (I × N) in (iv) can be
equivalently replaced by
CE∗ ran TΛ,I = H.
Indeed, assume that the first relation holds. As E is a frame for H, for any h ∈ H
there exists c ∈ `2 (I × N) such that CE∗ c = h. Now, c = c1 + c2 with c1 ∈ ran TΛ,I
and c2 ∈ ker CE∗ . Hence, h = CE∗ c1 ∈ CE∗ ran TΛ,I . Conversely, if CE∗ ran TΛ,I = H and
c ∈ `2 (I × N), then CE∗ c = CE∗ TΛ,I h for some h ∈ H. Thus, c − TΛ,I h ∈ ker CE∗ , i.e.,
c ∈ ran TΛ,I + ker CE∗ .
(d) If (ii) holds, then the range ran TΛ,I in (iv) is dense in `2 (I × N). Indeed, since (ii)
means that the operator TΛ is bounded and everywhere defined on H 2 (D) (cf. Theorem
2.10), the claim follows from Lemma 2.7.
(e) The condition (ii) of Theorem 3.8 cannot be strengthened by stating that (λj )j∈N
is a uniformly separated sequence itself. For a counterexample, let m := |I| and let
Λ = (λj )j∈N be a union of m uniformly separated subsequences Λi , i = 1, . . . , m, such
that λj 6= λk for j 6= k and ind(Λ) = m. Note that each Λi isPnecessarily infinite by
Corollary 2.12. For i = 1, . . . , m let Λi = (λj,i )j∈N and put Ai := ∞
j=0 λj,i h·, ej iej , where
(ej )j∈N is an orthonormal basis of a Hilbert space H. Choose vectors g1 , . . . , gm ∈ H
such that (Ani giL
)n∈N is a frame forLH, i = 1, . . . , m. This is possible by Theorem 3.4.
m
Now, put H := m
i=1 H and A :=
i=1 Ai , and let fi be the vector in H with (fi )i = gi
and (fi )k = 0 for k ∈ {1, . . . , m} \ {i}. Then (An fi )n∈N, i∈I is a frame for H, since for
h = h1 ⊕ . . . ⊕ hm ∈ H we have
m X
∞
m X
∞
X
X
|hh, An fi i|2 =
|hhi , Ani gi i|2 ,
i=1 n=0
i=1 n=0
17
from where it is easily seen that the claim is true.
Proof of Theorem 3.8. Suppose that A is a frame for H. By Theorem 3.1, the spectrum
of A is contained in D and E(T) = 0. Moreover, Lemma 3.2 shows that the spectrum
of A in D consists of infinitely many isolated eigenvalues
with finite multiplicities. By
P
Lemma 2.3, A is a diagonal operator. Let A = ∞
λ
P
be its normal form as in (i).
j
j
j=0
Then λj ∈ D and dim Pj H < ∞ for all j ∈ N. Let α, β > 0 be the frame bounds of A.
That is, we have
2
αkhk ≤
∞
XX
|hh, An fi i|2 ≤ βkhk2 ,
h ∈ H.
(3.6)
i∈I n=0
Fix j ∈ N. If h ∈ Pj H, then A∗ h = λj h and hence |hh, An fi i| = |h(A∗ )n h, fi i| =
|λj |n |hh, fi i|. Therefore,
P
∞ X
∞ X
X
X
|hh, fi i|2
n
2
2n
2
.
|hh, A fi i| =
|λj | |hh, fi i| = i∈I
1 − |λj |2
n=0 i∈I
n=0 i∈I
Together with (3.6), this proves (iii). From (iii) it follows that (Pj fi )i∈I is a frame for the
finite-dimensional space Pj H. Thus, |I| ≥ dim Pj H for each j ∈ N, which is equivalent
to mult(A) ≤ |I|. From (iii) we moreover conclude that
X
kPj fi k2 ≥ αε2j (dim Pj H) ≥ αε2j
(3.7)
i∈I
p
for each j ∈ N, where (cf. (3.3)) εj := 1 − |λj |2 , j ∈ N. For this, simply choose an
orthonormal basis of Pj H and plug its vectors into h in (3.5). Thus, for ϕ ∈ H 2 (D) we
obtain for the evaluation operator TΛ from (2.5) that
kTΛ ϕk22 =
∞
X
ε2j |ϕ(λj )|2 ≤ α−1
j=0
∞
XX
|ϕ(λj )|2 kPj fi k2 .
(3.8)
i∈I j=0
By Lemma 3.6, the latter expression is bounded from above by Ckϕk2H 2 (D) , where C is
some positive constant. Thus, the operator TΛ is everywhere defined and bounded. The
claim (ii) now follows from Theorem 2.10. To prove (iv), we start by noting that TΛ,I
2
is bounded
TΛ is. Let h ∈ H. Then there exists (cin )i∈I,
P Pas
P∞n∈N ∈ `n (I × N) such that
∞
n
2
h = i∈I n=0 cin A fi . Define ϕi ∈ H (D) by ϕi (z) = n=0 cin z for i ∈ I and put
φ := (ϕi )i∈I ∈ HI2 . Then
h=
∞
XX
i∈I j=0
Pj
∞
X
n=0
n
cin A fi =
∞ X
∞
XX
i∈I j=0 n=0
cin λnj Pj fi
=
∞
XX
εj ϕi (λj )(ε−1
j Pj fi ),
i∈I j=0
and thus h = CE∗ (TΛ ϕi )i∈I = CE∗ TΛ,I φ. Hence, we have CE∗ ran TΛ,I = H, which implies
(iv), see Remark 3.9(b).
Conversely, let the conditions (i)–(iv) be satisfied. Let us first prove that (An fi )n∈N
is a Bessel sequence for each i ∈ I. For this, fix i ∈ I and deduce from (iii) that
kPj fi k2 ≤ βε2j holds for each j ∈ N. Note that the evaluation operator TΛ from (2.5) is
18
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
everywhere defined and bounded by (ii) and Theorem 2.10. Thus, for every ϕ ∈ H 2 (D)
we have
∞
∞
X
X
|ϕ(λj )|2 kPj fi k2 ≤ β
|εj ϕ(λj )|2 = βkTΛ ϕk22 ≤ βkTΛ k2 kϕk2H 2 (D) .
j=0
j=0
Hence, the condition (iii) in Lemma 3.6 is satisfied so that (An fi )n∈N indeed is a Bessel
sequence. Let h ∈ H be arbitrary. By (iv) (see also Remark 3.9(b)), there exists φ ∈ HI2 ,
φ
(ϕi )i∈I , such that CE∗ TΛ,I φ = h. For i ∈ I, let (cin )n∈N ∈ `2 (N) such that ϕi (z) =
P=
∞
n
n=0 cin z for z ∈ D. Then
∞
XX
i∈I n=0
cin An fi =
∞ X
∞
XX
cin λnj Pj fi =
i∈I j=0 n=0
∞
XX
εj ϕi (λj )(ε−1
j Pj fi ) = h.
i∈I j=0
Hence, the synthesis operator of A =
frame for H.
(An fi )n∈N, i∈I
is onto, which means that A is a
From Theorem 3.8 we deduce the following two corollaries.
P
Corollary 3.10. Let A = ∞
j=0 λj Pj be a diagonal operator in normal form such that
(λj )j∈N is uniformly separated and let (fi )i∈I be a finite sequence of vectors in H. Then
A is a frame for H if and only if there exist α, β > 0 such that
X
α(1 − |λj |2 )khk2 ≤
|hh, fi i|2 ≤ β(1 − |λj |2 )khk2 , j ∈ N, h ∈ Pj H.
i∈I
Proof. Since (λj )j∈N is assumed to be uniformly separated, the conditions (ii) and (iv)
from Theorem 3.8 are satisfied (see Theorem 2.9). Thus, A is a frame for H if and only
if condition (iii) from Theorem 3.8 holds.
P∞
Corollary 3.11. Assume that A = j=0 λj Pj is a diagonal operator in normal form
and that dim Pj H = |I| < ∞ for all but a finite number of j ∈ N. Then A is a frame for
H if and only if the following two statements hold:
(a) (λj )j∈N is a uniformly separated sequence in D.
(b) There exist α, β > 0 such that for all j ∈ N and all h ∈ Pj H we have
X
α(1 − |λj |2 )khk2 ≤
|hh, fi i|2 ≤ β(1 − |λj |2 )khk2 .
i∈I
Proof. If (a) and (b) hold, then A is a frame for H by Corollary 3.10. Conversely, let
A be a frame for H. Then (b) follows from Theorem 3.8. For the proof of (a) let us
first assume that dim Pj H = |I| for all j ∈ N. Since (ε−1
j Pj fi )i∈I is a frame for Pj H
for every j ∈ N with frame bounds α and β (see Remark 3.9), we conclude that it is
even a Riesz basis of Pj H with Riesz bounds α and β. Hence, E = (ε−1
j Pj fi )j∈N, i∈I is
a Riesz basis of H. Thus, the synthesis operator CE∗ is one-to-one, i.e., ker CE∗ = {0}.
Therefore, it is a consequence of Theorem 3.8 that TΛ,I is onto. By definition of TΛ,I ,
also TΛ : H 2 (D) → `2 (N) is onto which is equivalent to (λj )j∈N being uniformly separated
(see Theorem 2.9).
P
For the general case, let J := {j ∈ N : dim Pj H = |I|} and let P := j∈J Pj . Then
P
A0 := A|P H = j∈J λj (Pj |P H) and (An0 P fi )n∈N, i∈I is a frame for P H. Thus, (λj )j∈J is
19
uniformly separated by the first part of the proof. The claim now follows from Corollary
2.12.
According
P to Theorem 3.8, if A is a frame for H, the operator A is a diagonal operator,
i.e., A = ∞
j=0 λj Pj , and the sequence of its (distinct) eigenvalues Λ = (λj )j∈N is a finite
union of uniformly separated sequences. It is therefore natural to ask about the index
ind(Λ) of Λ (see Definition 2.13), which is the minimum of those numbers n for which
there exists a partition into uniformly separated subsequences Λ1 , . . . , Λn of Λ. Note that
each separated subsequence of Λ is automatically uniformly separated by Theorem 2.11.
Theorem 3.12. Let A be a diagonal operator, let Λ = (λj )j∈N be the sequence of its
distinct eigenvalues, and let (fi )i∈I be a finite sequence of vectors in H. If A is
pa frame for
H with frame bounds α and β, then each pseudo-hyperbolic ball of radius r < α(8β|I|)−1
contains at most |I| elements of the sequence Λ. In particular, we have
ind(Λ) ≤ |I|.
(3.9)
Proof. First of all, note that if we can prove the main claim, the “in particular”-part
follows from Lemma 2.14. Let m := |I|. Towardspa contradiction, suppose that some ball
Br (z0 ) contains m + 1 elements of Λ, where r < α(8β|I|)−1 . Without loss of generality,
we may assume that these elements are λ1 , . . . , λm+1 . Since % is a metric on D, we have
%(λj , λk ) < 2r for all j, k = 1, . . . , m + 1.
m
Using the notation of Theorem 3.3, let ~yj := ~yj,1 = ε−1
j (hej,1 , fi i)i∈I ∈ C , j ∈ N. Since
~y1 , . . . , ~ym+1 are m + 1 vectors in Cm , there exists some c ∈ Cm+1 such that kck2 = 1
P
yj = 0. By Theorem 3.3, (~yj Kλj )j∈N is a Riesz sequence with Riesz bounds
and m+1
j=1 cj ~
α and β. In particular, we have k~yj k22 ≤ β for all j ∈ N. Using this, Cauchy-Schwarz,
and Lemma 2.8, we obtain
m+1
2
m
2
m
X
X
X
~yj (Kλ − Kλ ) 2 2
α≤
cj ~yj Kλj = cj ~yj Kλj − Kλm+1 ≤
j
m+1
HI
2 2
j=1
≤ 2β
m
X
HI
j=1
HI
j=1
%(λj , λm+1 )2 ≤ 8βmr2 < α,
j=1
which is the desired contradiction. Here, we also used that k~y ϕkH 2 = k~y k2 kϕkH 2 (D) for
I
~y ∈ Cm and ϕ ∈ H 2 (D), cf. (3.2).
Remark 3.13. (a) As a consequence of Theorem 3.12, one can add the condition that
ind(Λ) ≤ |I| to condition (ii) in Theorem 3.8 without changing the statement of the
theorem. If we do this, then Theorem 3.8 becomes a proper generalization of Theorem
3.4 because in the case |I| = 1 the new condition (ii) then states that Λ is uniformly
seperated, so that condition (iv) is automatically satisfied.
(b) The estimate ind(Λ) ≤ |I| from Theorem 3.12 is sharp in the sense that for every
given finite index set I there exist a diagonal operator A and vectors fi , i ∈ I, such that
(An fi )n∈N, i∈I is a frame for H and ind(Λ) = |I| (where Λ is the sequence of the distinct
eigenvalues of A). For an example, we refer the reader to Remark 3.9(e).
20
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
4. Dynamical Sampling with general bounded operators
In this section, we drop the requirement that A ∈ L(H) be normal. Similarly as before,
we shall fix A ∈ L(H), an at most countable index set I, and vectors fi ∈ H, i ∈ I, and
define
A := A(A, (fi )i∈I ) := (An fi )n∈N, i∈I .
Recall that an operator T ∈ L(H) is said to be strongly stable if T n f → 0 as n → ∞
holds for each f ∈ H. In this case, it follows from the Banach-Steinhaus theorem that T
is power-bounded, i.e., supn∈N kT n k < ∞. Consequently, we infer from Gelfand’s formula
for the spectral radius that r(T ) = sup{|λ| : λ ∈ σ(T )} = limn→∞ kT n k1/n ≤ 1. Hence,
the spectrum of a strongly stable operator is contained in the closed unit disk. The
following lemma was proved in [5] (see also (3.1)).
Lemma 4.1. If A is a frame for H, then A∗ is strongly stable. In particular, σ(A) ⊂ D.
Corollary 4.2. Assume that A is a frame for H. Then neither A nor A∗ has eigenvalues
on T. In particular,
σ(A) ∩ T ⊂ σc (A)
and
σ(A∗ ) ∩ T ⊂ σc (A∗ ).
(4.1)
Proof. Assume that λ ∈ T and f ∈ H are such that A∗ f = λf . Then kf k = kλn f k =
k(A∗ )n f k → 0 as n → ∞ by Lemma 4.1. Thus, f = 0. If Af = λf , then
kf k2 = |hλn f, f i| = |hAn f, f i| = |hf, (A∗ )n f i| → 0
as n → ∞. Again, f = 0 follows. The inclusions in (4.1) now simply follow from the fact
that ker T ∗ = (ran T )⊥ for T ∈ L(H).
The next proposition completes the necessary condition from Lemma 4.1 to a characterizing set of conditions.
Proposition 4.3. The system A = (An fi )n∈N, i∈I is a frame for H if and only if the
following conditions are satisfied.
(i) A∗ is strongly stable.
(ii) (fi )i∈I is a Bessel sequence.
(iii) There exists a boundedly invertible positive selfadjoint operator S ∈ L(H) such
that
ASA∗ = S − F,
(4.2)
where F is the frame operator of (fi )i∈I .
If the conditions (i)–(iii) are satisfied, then the operator S in (iii) is necessarily the frame
operator of A.
Proof. Assume that A is a frame for H and let S be its frame operator. As a subsequence
of the frame A, (fi )i∈I is a Bessel sequence. Furthermore, for f ∈ H we have
∞
∞
XX
XX
hA∗ f, An fi iAn+1 fi =
hf, An fi iAn fi
ASA∗ f =
=
i∈I n=0
∞
XX
hf, An fi iAn fi −
i∈I n=0
i∈I n=1
X
hf, fi ifi = Sf − F f,
i∈I
21
which proves (4.2). For the converse statement, assume that (i)–(iii) are satisfied. From
(iii) (and (ii)) we conclude that
A2 S(A∗ )2 = A(ASA∗ )A∗ = A(S − F )A∗ = ASA∗ − AF A∗ = S − (F + AF A∗ ).
For n ∈ N, n ≥ 1, we obtain by induction
An S(A∗ )n = S −
n−1
X
Ak F (A∗ )k = S −
k=0
n−1
XXD
E
· , Ak fi Ak fi .
k=0 i∈I
Hence, for f ∈ H and each n ∈ N we have
n−1
2
E2
X X D
f, Ak fi = hSf, f i − S 1/2 (A∗ )n f .
k=0 i∈I
Letting n → ∞ and using (i), we conclude from the conditions on S in (iii) that A is a
frame for H. Moreover, if S0 denotes the frame operator of A, then h(S − S0 )f, f i = 0
for f ∈ H, which implies S = S0 .
Although A itself might not be a contraction, the next proposition shows in particular
that it is at least similar to a contraction.
Corollary 4.4. If A is a frame for H with frame operator S, then
−1/2 1/2 S
AS ≤ 1.
Proof. Define B := S −1/2 AS 1/2 . From ASA∗ = S − F (see (4.2)) we infer that BB ∗ =
Id −F 0 , where F 0 = S −1/2 F S −1/2 . Note that F 0 is a non-negative selfadjoint operator
(since F is). Therefore, for f ∈ H we have
kB ∗ f k2 = hBB ∗ f, f i = kf k2 − hF 0 f, f i ≤ kf k2 ,
which shows that kBk ≤ 1.
In what follows, we will mainly focus on the situation in which I is a finite index set
– or at least the frame operator of (fi )i∈I is a compact operator. In this case, it is clear
that (fi )i∈I itself cannot be a frame for H unless dim H < ∞. The next proposition is
key to most of our observations below. Recall that an operator T ∈ L(H) is said to be
finite-dimensional or of finite-rank if dim ran T < ∞.
Proposition 4.5. Assume that A is a frame for H. If the frame operator F of (fi )i∈I is
compact, then for each λ ∈ D the operator A∗ − λ is upper semi-Fredholm. If |I| is finite
(in which case F is even finite-dimensional ), then
nul(A∗ − λ) ≤ |I|,
λ ∈ D.
(4.3)
Proof. We derive the claim from the identity (4.2). For λ ∈ D we have
AS(A∗ − λ) = ASA∗ − λAS = S − F − λAS = (Id −λA)S − F.
For all λ ∈ D the operator Bλ := Id −λA is boundedly invertible. This is clear for λ = 0,
and for λ 6= 0 we have Bλ = λ(λ−1 − A), which is boundedly invertible as σ(A) ⊂ D.
Thus,
Bλ−1 AS(A∗ − λ) = S − Bλ−1 F.
(4.4)
22
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
By Theorem 2.4, the operator on the right hand side is Fredholm, and so A∗ − λ is upper
semi-Fredholm by Lemma 2.5.
Now, let |I| be finite and let λ ∈ D be an eigenvalue of A∗ . If f is a corresponding
eigenvector, then (4.4) yields f = S −1 Bλ−1 F f . Hence, ker(A∗ − λ) ⊂ S −1 Bλ−1 ran F ,
which implies (4.3) as dim ran F ≤ |I|.
In the proof of the next theorem we heavily make use of the punctured neighborhood
theorem, Theorem 2.6.
Theorem 4.6. If A is a frame for H and the frame operator of (fi )i∈I is compact, then
ind(A∗ − λ) = ind(A∗ ) for each λ ∈ D and exactly one of the following cases holds:
(i) σ(A∗ ) = D and σp (A∗ ) = D.
(ii) σ(A∗ ) = D and σp (A∗ ) is discrete in D.
(iii) σ(A∗ ) is discrete in D.
In the case (i), each λ ∈ D is an eigenvalue of A∗ with infinite algebraic multiplicity,
whereas in the cases (ii) and (iii) the eigenvalues of A∗ in D have finite algebraic multiplicities. If case (iii) holds, then we have ind(A∗ ) = 0.
Proof. By Proposition 4.5, A∗ −λ is upper semi-Fredholm for each λ ∈ D. Hence it follows
from the punctured neighborhood theorem, Theorem 2.6, and a compactness argument
that ind(A∗ − λ) is constant on D. Similarly, one sees that nul(A∗ − λ) is constant on
D \ ∆, where ∆ is a discrete subset of D. Denote this constant value by n0 . Then it is
immediate that case (i) is satisfied exactly if n0 > 0. If n0 = 0, then case (iii) occurs if
and only if ind(A∗ ) = 0.
Let λ0 ∈ D be an eigenvalue of A∗ . For λ 6= λ0 close to λ0 we have λ ∈ D \ ∆ and
hence n0 = nul(A∗ − λ) = nul(A∗ − λ0 ) − k, where (see Theorem 2.6)
k = dim ker(A∗ − λ0 )/ (ker(A∗ − λ0 ) ∩ R∞ (A∗ − λ0 )) .
Hence, cases (ii) and (iii) occur exactly when nul(A∗ −λ0 ) = k. This happens if and only
if ker(A∗ − λ0 ) ∩ R∞ (A∗ − λ0 ) = {0}. But the latter means that the algebraic multiplicity
of λ0 as an eigenvalue of A∗ is finite.
The next corollary is an immediate consequence of Theorem 4.6.
Corollary 4.7. If dim H = ∞, A is a frame for H, and the frame operator of (fi )i∈I
is compact, then r(A) = 1. In particular, A is not uniformly stable (i.e., kAn k → 0 as
n → ∞).
Corollary 4.8. If dim H = ∞, A is a frame for H with frame operator S, and the frame
operator of (fi )i∈I is compact, then
−1/2 1/2 S
AS = 1.
Proof. By Corollary 4.4, B := S −1/2 AS 1/2 is a contraction. Since B is similar to A, we
have σ(B) = σ(A) and therefore, by Corollary 4.7, 1 = r(A) = r(B) ≤ kBk ≤ 1.
We define the essential spectrum σess (T ) of T ∈ L(H) by those λ ∈ C for which T − λ
is not semi-Fredholm.
23
Corollary 4.9. Assume that A is a frame for H and the frame operator of (fi )i∈I is
compact. Then
σess (A∗ ) = σc (A∗ ) = σ(A∗ ) ∩ T.
If, in addition, ind(A∗ ) 6= 0, then
σess (A∗ ) = σc (A∗ ) = T.
Proof. σc (A∗ ) ⊂ σess (A∗ ) holds by definition and σess (A∗ ) ⊂ σ(A∗ ) ∩ T is a direct consequence of Proposition 4.5. The remaining inclusion σ(A∗ ) ∩ T ⊂ σc (A∗ ) holds due to
Corollary 4.2. If ind(A∗ ) 6= 0, then either case (i) or case (ii) holds. In these cases, we
have σ(A∗ ) = D and hence, clearly, σ(A∗ ) ∩ T = T.
In the proof of Theorem 4.6 we have not used that the operator A∗ is strongly stable.
We incorporate this in the proof of the next theorem, where we make use of a theorem
from [18].
Theorem 4.10. Let I be finite and assume that A is a frame for H. Then case (ii)
is only possible if ind(A∗ ) = −∞. If ind(A∗ ) = 0, then case (iii) holds and also A is
strongly stable. In addition,
A∗ = S −1/2 U Id −R−1 F R−1 S 1/2 ,
(4.5)
where U is unitary and R boundedly invertible. In particular, A is similar to a finitedimensional perturbation of a unitary operator.
Proof. Let S and F be the frame operators of A and (fi )i∈I , respectively, and define
B := S −1/2 AS 1/2 . By Corollary 4.4, the operator B is a contraction. Moreover, also B ∗
is strongly stable and BB ∗ = I − F 0 , where F 0 = S −1/2 F S −1/2 . Since B ∗ and A∗ are
similar, it suffices to prove the corresponding statements for the operator B ∗ .
First, assume that ind(B ∗ ) > −∞ (note that ind(B ∗ ) < ∞, since nul(B ∗ ) < ∞). Let
∗
B = V |B ∗ | be the polar decomposition of B ∗ , where V is a partial isometry satisfying
ran V = ran B ∗ as well as ker V = ker B ∗ and |B ∗ | = (BB ∗ )1/2 (see, e.g., [8]). Thus, V
is Fredholm with ind(V ) = ind(B ∗ ). Since
F 0 = Id −BB ∗ = Id −|B ∗ |2 = (Id +|B ∗ |)1/2 (Id −|B ∗ |)(Id +|B ∗ |)1/2 ,
we have that
B ∗ = V |B ∗ | = V Id −(Id +|B ∗ |)−1/2 F 0 (Id +|B ∗ |)−1/2 = V Id −L−1 F 0 L−1 ,
(4.6)
where L := (Id +|B ∗ |)1/2 . That is, B ∗ = V − G, where G = V L−1 F 0 L−1 , and we
obtain B ∗ B = (V − G)(V ∗ − G∗ ) = V V ∗ − H, where dim ran H < ∞. Now, V V ∗ is
the orthogonal projection Pran V onto ran V = ran B ∗ , which, by assumption, is finitecodimensional. Therefore, B ∗ B = I − (I − Pran V ) − H = I − (Pker B + H), the operator
in parentheses being of finite-rank. Hence, [18, Theorem 2] applies to the operator B,
which states that σp (B) 6= D. Equivalently, we have def(B ∗ − λ) = nul(B − λ) = 0 for all
λ ∈ D \ ∆, where ∆ ⊂ D is at most discrete in D. In particular, case (ii) does not occur.
Now, assume that ind(B ∗ ) = 0. From what we just proved, it follows that also nul(B ∗ −
λ) = 0 for all λ ∈ D \ ∆. Thus, σ(B ∗ ) ∩ D ⊂ ∆ is discrete in D, meaning that case (iii)
holds. To conclude the proof, let W be a partial isometry with ker W = ran B and
24
C. CABRELLI, U. MOLTER, V. PATERNOSTRO, AND F. PHILIPP
ran W = ker B (which exists due to ind(B ∗ ) = 0, cf. [8, Lemma 2.9]). Then U := V + W
is unitary and B ∗ = U |B ∗ |, so that we may replace V in (4.6) by U . The formula (4.5)
follows from here by a similarity transformation (and R = S 1/2 L). The fact that also B
is strongly stable is now an immediate consequence of [18, Corollary 3].
The following example shows that case (i) in Theorem 4.6 cannot be neglected as a
possibility for an operator generating a frame.
Example 4.11. Let H = `2 (N) and let A be the right-shift, i.e., (Af )(0) = 0 and
(Af )(j) = f (j − 1) for j ≥ 1, f ∈ `2 (N). Moreover, let f0 ∈ `2 (N) be defined by
f0 (j) = δj,0 , j ∈ N. Thus, f0 is the first standard unit vector. It is easy to see that
(An f0 )n∈N is the standard ONB of `2 (N) and therefore a frame for H. The left-shift A∗
is surjective and has a one-dimensional kernel. In fact, for all λ ∈ D the operator A∗ − λ
has this property. Hence, every λ ∈ D is an eigenvalue of A∗ . The algebraic multiplicity
of each λ ∈ D is infinite since ran(A∗ − λ) = H.
Recall that a semigroup of operators is a collection (Tt )t≥0 ⊂ L(H) satisfying Ts+t =
Ts Tt for all s, t ∈ [0, ∞).
Proposition 4.12. Let A∗ = Tt0 , t0 > 0, be an instance of a semigroup (Tt )t≥0 of
operators and let the frame operator of (fi )i∈I be compact. Then if A is a frame for H,
either case (ii) or case (iii) occurs. In case (ii) we have ind(A∗ ) = −∞.
m = A for each m. Let λ ∈ D
Proof. Let Bm := Tt∗0 /m , m ∈ N, m ≥ 1. Then we have Bm
∗ )m − λm = P (B ∗ , λ)(B ∗ − λ), where P (B ∗ , λ) is
be arbitrary. Then A∗ − λm = (Bm
m
m
m
∗
∗ − λ is upper semia polynomial in Bm and λ. This and Lemma 2.5 imply that Bm
∗ ). In
Fredholm. Moreover, by the index formula (2.4) we have that ind(A∗ ) = m · ind(Bm
particular, ind(A∗ ) is divisible by each m ∈ N, m ≥ 2. Thus, ind(A∗ ) ∈ {0, −∞}. Note
that ind(A∗ ) = +∞ is not possible since nul(A∗ ) < ∞.
Now,
let λ ∈ D \ {0}, λ = reit , r ∈ (0, 1), t ∈ [0, 2π), let n = nul(A∗ − λ) + 1, and put
√
n
λk := r exp( ni (t + 2kπ)), k = 0, . . . , n − 1. Suppose that each λk is an eigenvalue of Bn∗
with eigenvector gk , k = 0, . . . , n − 1. Then each gk is an eigenvector of A∗ with respect
to λ as A∗ gk = (Bn∗ )n gk = λnk gk = λgk . But as the gk are linearly independent, this is
a contradiction to the choice of n. Thus, the eigenvalues of Bn∗ do not fill the open unit
disk. In turn, σp (Bn∗ ) is discrete in D and hence the same holds for σp (A∗ ).
Note that we have not used any continuity properties of the semigroup in the proof
above. In fact, if (Tt )t≥0 is a C0 -semigroup, then it can be shown that under the conditions
of Proposition 4.12, ker(Tt ) = {0} for each t ≥ 0, which in particular excludes case (i).
Theorem 4.13. Let I be finite. In order that A be a Riesz basis of H it is necessary
that A∗ satisfies case (i).
Proof. Let A be a Riesz basis of H. For n ∈ N, n ≥ 1, define the two closed subspaces
n
o
n
o
Mn,− := span Ak fi : i ∈ I, 0 ≤ k < n
and Mn,+ := span Ak fi : i ∈ I, k ≥ n .
As A is a Riesz basis, we have that
Mn,− u Mn,+ = H.
25
Since
⊥
f ∈ Mn,+
⇐⇒ hf, An+k fi i = 0 ∀k ≥ 0, i ∈ I ⇐⇒ (A∗ )n f = 0,
⊥ = ker((A∗ )n ). Hence, we have
we conclude that Mn,+
⊥
dim ker((A∗ )n ) = dim Mn,+
= codim Mn,+ = dim Mn,− = n|I|.
Consequently, the algebraic multiplicity of the eigenvalue λ = 0 of A∗ is infinite. By
Theorem 4.6 this only happens if case (i) is satisfied.
Proposition 4.12 and Theorem 4.13 immediately yield the following corollary.
Corollary 4.14. Let A∗ = Tt0 , t0 > 0, be an instance of a semigroup T of operators.
Then, for any finite sequence (fi )i∈I of vectors in H, the system (An fi )n∈N, i∈I is never
a Riesz basis of H.
Acknowledgements. The research of the authors was partially supported by UBACyT
under grants 20020130100403BA, 20020130100422BA, and 20020150200110BA; CONICET (PIP 11220110101018), and MinCyT Argentina under grant PICT-2014-1480.
Moreover, the authors would like to thank D. Suárez for sharing his knowledge on sequences in the unit disk.
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(C. Cabrelli, U. Molter and V. Paternostro) Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Matemática, Buenos Aires, Argentina, and
CONICET-Universidad de Buenos Aires, Instituto de Investigaciones Matemáticas Luis A.
Santalo (IMAS), Buenos Aires, Argentina
E-mail address, C. Cabrelli: [email protected]
URL, C. Cabrelli: http://mate.dm.uba.ar/~cabrelli/
E-mail address, U. Molter: [email protected]
URL, U. Molter: http://mate.dm.uba.ar/~umolter/
E-mail address, V. Paternostro: [email protected]
(F. Philipp) Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Departamento de Matemática, Buenos Aires, Argentina
E-mail address, F. Philipp: [email protected]
URL, F. Philipp: http://cms.dm.uba.ar/members/fmphilipp/