ESSENTIAL CODIMENSION AND DIAGONALS
OF FINITE SPECTRUM NORMAL OPERATORS
JIREH LOREAUX
Abstract. Kadison characterized the diagonals of projections and observed
the presence of an integer, which Arveson later recognized as a Fredholm index
obstruction applicable to any normal operator with finite spectrum coincident
with its essential spectrum whose elements are the vertices of a convex polygon.
Recently, in joint work with Kaftal, the author linked the Kadison integer to
essential codimension of projections. This paper provides an analogous link
between Arveson’s obstruction and essential codimension.
∞
A diagonal of an operator T ∈ B(H) is a sequence hT en , en i n=1 where {en }∞
n=1
is an orthonormal basis. In other words, a diagonal of T is the diagonal of some
matrix representation for T with respect to an orthonormal basis.
In his seminal paper [Kad02] Kadison proved the following characterization of
diagonals of projections which he referred to as the Carpenter’s and Pythagorean
Theorems.
Theorem 1 (Kadison). A sequence (dn )∞
n=1 with values in the unit interval is the
diagonal of a projection if and only if the quantities
X
X
a :=
dn and b :=
(1 − dn )
dn ≥1/2
dn <1/2
satisfy one of the mutually exclusive conditions
(1) a + b = ∞;
(2) a + b < ∞ and a − b ∈ Z.
The reader should take note that there is nothing particularly special about
the value of 1/2 in the above theorem. Indeed, we may replace it with any value
α ∈ (0, 1). However, the integer a − b is of special interest. When P is a finite
projection this integer makes perfect sense because b is a finite sum and so a − b is
Tr(P ) less a finite number of 1’s, and is therefore an integer. Similarly for cofinite
projections, just apply the previous argument to P ⊥ and swap the roles of a, b. But
for infinite, coinfinite projections the existence of this integer is not at all obvious
and Kadison himself referred to it as “curious.”
Arveson soon recognized this integer as the index of a Fredholm operator in
[Arv07], and referred to it as an “index obstruction” to an arbitrary sequence
with values in the unit interval being a diagonal of a projection. Arveson was
able to extend this index obstruction to any normal operator with finite spectrum
coincident with its essential spectrum whose elements are the vertices of a convex
polygon.
Key words and phrases. Essential codimension, diagonals of projections, diagonals of normal
operators.
1
2
JIREH LOREAUX
Our main theorem is Theorem 7, which is a restatement of Arveson’s theorem
(see Theorem 4) where we drop the hypothesis on the essential spectrum and we
make explicit the Fredholm nature of Arveson’s index obstruction using essential
codimension. Moreover, our proof is significantly shorter than the one provided
by Arveson, using only a straightforward geometric result (Lemma 8) and a more
detailed version of Kadison’s theorem (Theorem 7) recently obtained by the author
and V. Kaftal in [KL17] which links the integer a − b to the essential codimension
of projections.
In order to state his main theorem, Arveson introduced several definitions which
we reproduce below for clarity.
Definition 2 (Arveson). Given a finite set {λ1 , . . . , λn } = X ⊂ C, define the
sequences which accumulate summably at X by
Lim1 (X) := (dn ) ∈ `∞ dist(dn , X) ∈ `1 .
The additive subgroup KX ⊆ C is the countable group whose elements take the
form
c1 λ1 + · · · + cn λn ,
where cj ∈ Z and c1 + · · · + cn = 0. The quotient ΓX := C/KX is called the
obstruction group.
The next definition is a slight deviation from the one Arveson provided. We
explain the distinction below.
Definition 3. Given a finite set X ⊂ C and (dn ) ∈ Lim1 (X), a renormalized sum
of (dn ) is an absolutely summable series of the form
∞
X
(dn − xn ),
n=1
where xn ∈ X.
Arveson proved that any two renormalized sums of (dn ) differ by an element of
KX , and therefore any element of Lim1 (X) uniquely identifies an element of the
obstruction group ΓX . In fact, it was actually this element of ΓX which he referred
to as the renormalized sum, but for our purposes it will be more useful to look at
the elements in C. Arveson also remarked that this natural map s : Lim1 (X) → ΓX
from summably accumulating sequences to their equivalence class of renormalized
sums is surjective.
With this terminology, we can state Arveson’s main theorem [Arv07, Theorem 4].
Pm
Theorem 4 (Arveson). Let N :=
k=1 λk Pk be a normal operator with finite
spectrum X = {λ1 , . . . , λm } coincident with its essential spectrum whose elements
1
are the vertices of a convex polygon in C. Then any diagonal
P∞(dn ) ∈ Lim (X)
of N has renormalized sums in the group KX . That is, if
n=1 (dn − xn ) is a
renormalized sum, then there exist integers c1 , . . . , cm which sum to zero for which
∞
m
X
X
(dn − xn ) =
ck λk .
n=1
k=1
In order to understand the operator-theoretic explanation of the integers cj in
Arveson’s theorem, we must recall the notion of essential codimension which arose
from the BDF theory [BDF73, Remark 4.9].
ESSENTIAL CODIMENSION AND FINITE SPECTRUM NORMAL OPERATORS
3
Definition 5 (Brown, Douglas and Fillmore). Given a pair of projections P, Q
whose difference is compact, the essential codimension of P in Q, denoted [P : Q],
is the integer defined by


Tr(P ) − Tr(Q) if Tr(P ), Tr(Q) < ∞,
[P : Q] :=
if Tr(P ) = Tr(Q) = ∞, where

ind(V ∗ W )
W ∗ W = V ∗ V = I, W W ∗ = P, V V ∗ = Q.
It is straightforward to show that under the above conditions V ∗ W is Fredholm
(in fact, it’s image in the Calkin algebra is unitary), and independent of the choice
of isometries W, V implementing the Murray–von Neumann equivalence between
the identity and P, Q, respectively. Equivalently, essential codimension may be
defined as the index of QP : P H → QH, which is Fredholm if P − Q is compact.
When P, Q are infinite, this may be realized by noting that V ∗ W = V ∗ QP W and
V ∗ , W are unitaries between QH → H and H → P H, respectively, but this index
formula ind(QP ) = [P : Q] also holds when P, Q are finite.
The link between essential codimension and the Kadison integer is expressed
in the following theorem [KL17, Proposition 2.8] which says that the integer is
precisely the essential codimension between the original projection and a natural
diagonal projection.
Theorem 6 (Kaftal–Loreaux). Suppose P, Q are projections. Then P − Q is
Hilbert–Schmidt if and only if in some (equivalently, every) orthonormal basis
{en }∞
n=1 which diagonalizes Q, the diagonal (dn ) of P satisfies a + b < ∞, where
X
X
a :=
dn and b :=
(1 − dn ).
en ∈QH
en ∈Q⊥ H
Moreover, in this case a − b = [P : Q].
This theorem identifies the integer in Kadison’s theorem in the following manner.
If P is a projection with diagonal (dn ) and a, b are as in Theorem 1 with a + b < ∞,
then, by choosing Q to be the projection onto span{en | dn ≥ 1/2}, Theorem 6
guarantees P −Q is Hilbert–Schmidt, a fact which was known to Arveson. Moreover,
this theorem establishes a − b = [P : Q].
We now state our main theorem which identifies the integers in Arveson’s Theorem 4 as the essential codimensions between the spectral projections of N and
diagonal projections arising naturally from the renormalized sum.
Pm
Theorem 7. Let N := k=1 λk Pk be a normal operator with finite spectrum X :=
{λ1 , . . . , λm } ⊂ C which are the vertices of a convex polygon and corresponding
spectral projections
P∞ P1 , . . . , Pm , and suppose (dn ) is a diagonal of N . If (dn ) ∈
Lim1 (X) and
n=1 (dn − xn ) is a renormalized sum, then Pk − Qk is Hilbert–
Schmidt, where Qk is the diagonal projection onto span{en | xn = λk }. Moreover,
∞
m
X
X
(dn − xn ) =
[Pk : Qk ]λk ,
n=1
and this sum lies in KX , i.e.,
k=1
Pm
k=1 [Pk
: Qk ] = 0.
In Arveson’s proof of Theorem 4 he also constructed some diagonal projections
Qk which sum to the identity and whose difference with Pk is Hilbert–Schmidt.
However, his Qk projections did not arise directly from the choices xn ∈ X in the
renormalized sum, which obfuscated the origin of these integers.
4
JIREH LOREAUX
One other interesting thing to note
is that there is a Fredholm operator of index
Pm
zero lurking about, namely T :=
k=1 Qk Pk , which is Fredholm because each
Q
P
is
Fredholm
as
an
operator
from
Pk H → Qk H. It’s index is ind(T ) =
k
k
Pm
[P
:
Q
]
=
0,
but
this
may
also
be
seen
by noting that
k
k
k=1
PnT is a Hilbert–Schmidt
Pn
perturbation of the identity. Write the identity as I =
j=1 Qk
j=1 Pj and
note that all terms Qj Pk are Hilbert–Schmidt for j 6= k since
(1)
Qj Pk = Qj (I − Pj )Pk = Qj (Qj − Pj )Pk .
Before we prove Theorem 7 we need a straightforward geometric lemma which
is in the same vein as [Arv07, Lemma 2]. We only prove this version to make our
proof of Theorem 7 cleaner and more self-contained.
Lemma
Pm 8. Let λ1 , . . . , λm ∈ C form the vertices of a convex polygon and suppose
x = j=1 cj λj is a convex combination of the vertices and let L be the line through
the vertices adjacent to λk . If x lies on a line parallel to L separating λk from the
remaining vertices, then
m
X
|x − λk |
cj ≤
.
dist(λ
k , L)
j=1
j6=k
Proof. Relabel the vertices if necessary so that k = 1. By applying an affine
transformation we may suppose that λ1 = 1 and L = −a + iR for some a ≥ 0 so
that <(x) = 0. Note that −a = maxj≥2 {<(λj )}. Since 0 ∈ [−a, 1] we may write
1
t max{<(λj )} + (1 − t)λ1 = 0, for t =
j≥2
1+a
Now
!
m
m
X
X
0 = <(x) =
cj <(λj ) ≤
cj max{<(λj )} + c1 λ1 .
j=1
j=2
j≥2
Since we have two convex combinations of −a, 1 and the latter is closer to 1 than
the former, the convexity coefficients satisfy
m
X
dist(<(x), λ1 )
|x − λ1 |
1
=
≤
.
cj ≤ t =
1
+
a
dist(λ
,
L)
dist(λ
1
1 , L)
j=2
We will also need one basic property of essential codimension, namely, if P −
Q, P 0 − Q0 are compact and P, P 0 and Q, Q0 are orthogonal pairs, then
(2)
[P : Q] + [P 0 : Q0 ] = [P + P 0 : Q + Q0 ].
This is seen most easily using the index representation ind(QP ) of essential codimension. First note that QP 0 and Q0 P are compact (see (1) for the argument). In
the display below, the first two Fredholm indices are taken with domain/codomain
pairs P H → QH and P 0 H → Q0 H, respectively, and the rest with (P + P 0 )H →
(Q + Q0 )H. We have
ind(QP ) + ind(Q0 P 0 ) = ind(QP + Q0 P 0 ) = ind (Q + Q0 )(P + P 0 ) .
Proof of Theorem 7. Let Λj = {n ∈ N | xn = λj }, so thatP
Qj is the projection onto
m
span{en }n∈Λj . Now, for each n ∈ N, we may write en = j=1 Pj en and therefore
dn = hN en , en i =
m
X
j=1
hPj en , en i λj ,
REFERENCES
5
which is a convex combination of the spectrum. Let L be the line through the two
vertices of X adjacent to λk . Since (dn ) ∈ Lim1 (X) we know that (dn − λk )n∈Λk
is absolutely summable. Therefore, for all but finitely many indices n ∈ Λk , the
diagonal entry dn lies on a line parallel to L separating λk from X \ {λk }. By
Lemma 8, for these indices
m
X
|dn − λk |
|dn − λk |
and hence hPk en , en i ≥ 1 −
.
hPj en , en i ≤
dist(λk , L)
dist(λk , L)
j=1
j6=k
Since the above display holds for all but finitely many n ∈ Λk and (dn − λk )n∈Λk
is absolutely summable, this proves hPj en , en i n∈Λ lies in Lim1 ({0}) = `1 when
k
j 6= k and in Lim1 ({1}) when j = k. Therefore, by Theorem 6, Pk − Qk is Hilbert–
Schmidt and
X
X
[Pk : Qk ] =
hPk en , en i +
hPk en , en i − 1 ,
n∈Λk
n∈N\Λk
where each sum is absolutely convergent. Multiplying by λk and summing over k,
m X
∞
m
X
X
X
(dn − xn )
[Pk : Qk ]λk =
hN en , en i − λk =
n=1
k=1 n∈Λk
k=1
as desired. Moreover, by (2) we have
"m
#
m
m
X
X
X
[Pk : Qk ] =
Pk :
Qk = [I : I] = 0,
k=1
k=1
k=1
so the renormalized sum lies in KX .
References
[Arv07]
[BDF73]
[Kad02]
[KL17]
W. Arveson, Diagonals of normal operators with finite spectrum, Proceedings
of the National Academy of Sciences of the United States of America 104.4
(2007), pp. 1152–1158, doi: 10.1073/pnas.0605367104.
L. G. Brown, R. G. Douglas, and P. A. Fillmore, Unitary equivalence modulo the
compact operators and extensions of C ∗ -algebras, Lecture Notes in Mathematics
vol. 345 (Apr. 13, 1973), ed. by P. A. Fillmore, pp. 58–128.
R. V. Kadison, The Pythagorean Theorem II: the infinite discrete case, Proceedings of the National Academy of Sciences of the United States of America
99.8 (2002), pp. 5217–5222, doi: 10.1073/pnas.032677299.
V. Kaftal and J. Loreaux, Kadison’s Pythagorean Theorem and essential codimension, Integral Equations and Operator Theory (2017), arXiv: 1609.06754
[math.OA], forthcoming.
E-mail address: [email protected]
Southern Illinois University Edwardsville, Department of Mathematics and Statistics, Edwardsville, IL, 62026-1653, USA