151
Bulletin of Society for mathematical services & standards (B SO MA S S)
Vol. I No. 2 (2012), pp. 151-171
ISSN: 2277- 8020
© www.ijmsea.com www.jsomass.com
ON SIMILARITY AND QUASISIMILARITY EQUIVALENCE
RELATIONS
Isaiah Nalianya Sitati
School of Mathematics, University of Nairobi, Chiromo Campus,
P. O. Box 30197-00100, Nairobi.
[email protected]
Musundi Sammy Wabomba
Chuka University College P.O. Box 109-60400, Kenya.
[email protected], [email protected]
Benard Mutuku Nzimbi
School of Mathematics, University of Nairobi, Chiromo Campus,
P. O. Box 30197-00100, Nairobi.
[email protected] , [email protected],
Kirimi Jacob
Chuka University College P.O. Box 109-60400, Kenya.
[email protected]
Abstract
Similarity and unitary equivalence can be shown to be of equivalence relations. We discuss a result
showing that two similar operators have equal spectra (i.e. point and approximate point spectrum). More
so, unitary equivalence results for invariant subspaces and normal operators are proved. For similar
normal operators, we state the Fuglede – Putnam –Rosenblum theorem that makes proofs for similar
normal operators more simplified. It is also noted that direct sums and summands are preserved under
unitary equivalence. Furthermore, we show that the natural concept of equivalence between Hilbert
Space operators is unitary equivalence which is stronger than similarity.
By introducing the notion of quasisimilarity of operators which is the same as similarity in finite
dimensional spaces, but in infinite dimensional spaces, it is a much weaker relation, we further show that
quasisimilarity is an equivalence relation. We also link invariant subspaces and hyperinvariant subspaces
with quasisimilarity where it is seen that similarity preserves nontrivial invariant subspaces while
quasisimilarity preserves nontrivial hyperinvariant subspaces.
Mathematics Subject Classification: 47A10; 47A15; 47B15
KEY WORDS: Similarity, equivalence relation, quasisimilarity.
1.1
In this paper, Hilbert spaces or subspaces will be denoted by capital letters
and
etc
, , , etc denote bounded linear operators where an operator means a bounded linear
transformation (equivalently, a continuous linear transformation) :
bounded linear transformations from
to
For an operator
,
and kernel of
Similarly,
, we denote by
.
denotes the set of
, which is equipped with the (induced uniform) norm.
,
, Ran
, Ker
, the spectrum, adjoint, norm, range
respectively.
and will denote the zero operator and identity operator on
An operator
an involution if
respectively.
is said to be:
=
self adjoint or Hermitian if
ℝ ∀
equivalently, if
unitary if
normal if
∀
(equivalently, if
),
an isometry if
a partial isometry if
or equivalently, if
is a projection.
From the above definitions we have the following proper class inclusion relations:
Unitary ⊆ Isometry ⊆ Partial isometry and Unitary ⊆Normal .
An operator
is said to be a scalar if it is a scalar multiple of the identity operator
,
152
left shift operator if
i.e.,
…..) ∀
…..) = (
given by
…..) and
where
is
, this operator is also called a backward
weighted shift operator with weights 1,
right shift operator if
…..) and
where
, also called the
unilateral shift or forward weighted shift operator with weights 1.
If
and
are Hilbert spaces, then their (orthogonal) direct sum will be denoted by
⊕ , which itself
is a Hilbert space.
A contraction is an operator such that
operator
with
(i.e.
, Ker ( ) is the set
}
is the set
subspace of
. A strict contraction is an
. The Kernel (or null space) of
Ker ( ) = ( ) ={
The range of
∀
(i.e.
={
}.
i.e, it is a linear manifold that is closed in H for every T
manifold that is not necessarily closed in
. An operator
and
is a
is a linear
is finite-dimensional if
is finite-
dimensional.
A set
in
is invariant for
⊆
if
.
is an invariant subspace for
which, as a subset of , is invariant for . A subspace
of
if it is a subspace of
is invariant for
if and only if
is
invariant for .
A subspace
reduces
(equivalently, if
(or
is a reducing subspace for ) if both M and
is invariant for both
is a hyperinvariant subspace for
If
is an invariant subspace for
=*
where
Let {
{
+
for operators
and
are invariant under
).
if it is invariant for every operator that commutes with it.
then, relative to the decomposition H=M⊕
:
is the restriction of
and
,
can be written as
,
on
} be a countable collection Hilbert spaces and consider the Hilbert space
} is a bounded set of operators (i.e.,
⊕
. Suppose
. The (orthogonal) direct sum of
153
{ } is the operator
such that
for each . This is denoted by
⊕
Each
reduces
and
. A direct summand of an operator is a restriction of it to
a reducing subspace. (E.g.
Let
is a direct summand of
be a Hilbert space and
. The set
is invertible is called the resolvent set of
of all complex number
for which
. Equivalently,
{ }
{
The complement of the resolvent set
}
denoted by
is called the spectrum of
{ }
{
such that
⊕
i.e,
} which is the set of all
fails to be invertible (i.e. fails to have a bounded inverse on
On the basis of this failure, the spectrum can be split into many disjoint parts. A classical disjoint
partition comprises of three parts: the set of those
is called the point spectrum of,
that
has no inverse, denoted by
i.e,
{ } , which is exactly the set of all eigenvalues of
{
A scalar
such that
is an eigenvalue of an operator
if there exists a nonzero vector
{ } . The set of all those
; equivalently, if
.
has a densely defined but unbounded inverse on its range, denoted by
such
for which
is called the continuous
spectrum of , i.e,
{ }
{
If
}
has an inverse that is not densely defined then
denoted
The parts
. That is
{ }
{
,
belongs to the residual spectrum of ,
}
are pairwise disjoint and
Other subsets of the spectrum exist such as: approximate point spectrum, denoted
belongs to the approximate point spectrum of
vectors {
Let
and
} such that
if and only if there exists a sequence of unit
.
be Hilbert spaces. An operator
is invertible if it is injective (one -to- one)
and surjective (onto or has dense range); equivalently if
the class of invertible linear operators by
; a number
.
154
{ } and
We denote
The commutator of two operators
and
is [
The self- commutator of an operator
Two operators
and
] is defined by
if there exists an operator
i.e,
where
which is an invertible operator from
Linear operators
and
unitary operator
.
]
are similar (denoted
such that
subalgebra of
, denoted by [
is a Banach
to
are unitarily equivalent (denoted
), if there exists a
such that
.
Two operators are considered the “same” if they are unitarily equivalent since they have the same,
properties of invertibility, normality, spectral picture (norm, spectrum and spectral radius).
An operator
is quasi-invertible or a quasi-affinity if it is an injective operator with dense
{ } and
range (i.e.
{ ̅ } and,
; equivalently,
is quasi-invertible if and only if
is quasi-invertible).
is a quasi –affine transform of
An operator
such that
if there exists a quasi-invertible
.
exists a quasi-invertible operator intertwining
Two operators
and
is a quasiaffine transform of
are quasi-similar (denoted T S) if they are quasi-affine
such that
and
).
is said to be densely intertwined to
(T) of
if there
to .
transforms of each other (i.e., if there exists quasi-invertible operators
The multiplicity
{ ̅ } thus
if there exists an operator with dense range intertwining
is the minimum cardinality of a set
such that
⋁
For the unilateral shift operator
and for the backward shift,
=
155
to .
1.2
A natural method for constructing an invariant subspace for an operator on a Hilbert space is to find a
second, known operator which is similar in some weak sense to the given operator and then use this
second operator and the weak similarity to construct the desired subspace. One such weak similarity is
the notion of quasisimilarity introduced by Sz.-Nagy and Foias [13].
T.B. Hoover [6] studied hyperinvariant subspaces and proved the result that if
operators acting on the Hilbert spaces
so does . If in addition,
and
respectively, and if
and
are quasi similar
has a hyperinvariant subspace, then
is normal, then the lattice of hyperinvariant subspaces for
lattice which is lattice isomorphic to the lattice of spectral projections for
contains a sub
Similar results for
hyperinvance have been studied by C.S. Kubrusly [9] and has show that similarity preserves nontrivial
subspaces while quasisimilarity preserves hyperinvariant subspaces. Hoover [1] has also shown some
properties of operators that are preserved by quasisimilarity and those that are not. He gave a result to
show that quasisimilar normal operators are unitarily equivalent. Quasisimilarity and weak similarity are
also discussed. In addition, quasisimilar isometries are shown to be unitarily equivalent. He also gave an
example to show that quasisimilarity preserves neither spectra nor compactness.
Equality of spectra of quasisimilar normal operators was proved in [2, Lemma 4.1 p. 683] but the result
did not generalize to pairs of quasisimilar hyponormal operators as seen in [4, solution156, p. 309] and [7,
Theorem 2.5. p.681]. In spite of these examples, Clary [1, Theorem 2, p.89] went ahead to show that
quasisimilar hyponormal operators had equal spectra. In 1981, J.G. stampfli [12] extended Clary’s results
on quasisimilar hyponormals where an inclusion relation for the spectra of quasisimilar operators
satisfying Dunford’s condition was obtained. It is in this paper that quasimilarity and the unilateral shift
was discussed. These results were extensively analyzed by Lambert [10] in his thesis where he showed
that any two unilateral weighted shifts which are quasisimilar are actually similar, but if unilateral shift is
replaced by bilateral as pointed out by Fialkow [3] the results fail.
A question on whether quasisimilar hyponormal operators had equal spectra as asked by S.Clary in [1]
was studied and answered by L.R.Williams in [17]. i.e. they indeed had equal spectra. He went further to
give some results that relate quasisimilarity and hyponormal operators; for instance if one of the quasiaffinities of the two hyponormals is compact, then they have equal essential spectra [15, Theorem A,
P.126].
156
By considering the natural concept of equivalence between Hilbert Space operators to be unitary
equivalence, we show that similarity is an equivalence relation on
Similarly by introducing the
notion of quasisimilarity of operators which is the same as similarity in finite dimensional spaces, but in
infinite dimensional spaces, it is a much weaker relation, we show that quasisimilarity is an equivalence
relation. Furthermore, we establish the link between invariant subspaces and hyperinvariant subspaces
with similarity and quasisimilarity operator properties.
We recall that two operators
and
an invertible operator
Similarly,
are similar (denoted by
) if there exists
such that
and
operator
.
are unitarily equivalent (denoted
) if there exisits a unitary
such that
.
Theorem 2.1. Unitary equivalence is an equivalence relation.
Proof: We show that unitary equivalence is (i) Reflective, (ii) Symmetric and (iii) Transitive.
(i)
Reflexivity: i.e.
Let
. Then
where
is a unitary operator. Hence
In this case, we may choose (without loss of generality
(ii)
Symmetry: i.e.
.
Now suppose that
. We show that
operator
).
. Let
and
. There exists a unity
such that
………………………………………………..
Pre multiplying
by
and post multiplying the same by
Hence
(iii) Transitivity i.e. If
exist unitary operators
.
gives
.
and
, then
and
. Suppose
and
. Then there
where is a Hilbert space such that
and
Using
and
we have that
unitary operator since
where
are unitary. Hence
157
.
is a
Remark 2.2. Similarly, it can be proved (as Theorem 2.1.1 above) that similarity is an equivalence
relation on
The natural concept of equivalence between Hilbert space operators is unitary
equivalence which is stronger than similarity.
A nice approach of determining whether two operators are similar or not is by use of matrices. We need to
compute a particular canonical form, and in our case, we may use the triangular or Jordan form which we
describe briefly as follows:
Definition 2.3 [23]. A matrix is said to be triangular if all its entries above the main diagonal are 0, or
equivalently, if
is a linear operator on
over a scalar field
, the matrix of
is in the basis
is triangular if
i.e, if
is a linear combination only of
and its predecessors in the basis.
Theorem 2.4 (Schur’s Theorem ). If an operator (matrix)
there exists an operator
such that
has all its eigenvalues in
is a triangular matrix.
Remark 2.5. Theorem 2.4 above is usually described by saying that
triangular form over
can be brought to a
In other words every matrix A on a finite dimensional space is unitarily
equivalent to a triangular matrix.
The elements of the main diagonal of its matrix are precisely the eigenvalues of .
 1
0 






0
Definition 2.6 [23]. (Jordan form). An m m matrix
with
then
on the main diagonal,
0
1

1

0




1 0
 1

0 
above the main diagonal and 0’s elsewhere such that
[ ]
 1
0 






{ 0
0
1

1

158
0




1 0
 1

0 
is a basic Jordan block belonging to
Corollary 2.7 [5]. Two linear operators in
which have all their eigenvalues in
are similar if
and only if they can be brought to the same Jordan form.
Remark 2.8. The Jordan form acts as a “determiner” for similarity classes to this type of linear operators.
Theorem 2.9. [5]. Let
be a linear operator on the complex
{ }
Then there is a basis
dimensional vector space
such that
[ ]
where
where
,there exists a nonsingular matrix
such that
Remark 2.10. The Jordan form of an operator or matrix plays a very important role in understanding the
structure of matrices and linear operators (very applicable in the study of differential equations).
Example 2.11.
Let
on
*
+ and
by
√
(
√
√
(√
√
√
i.e.
+ be two dimensional operators on
) whose inverse is
) (√
(
Then
*
)
(√
√
)(
√
√
)
)
(
√
).
(
) and
√
(√
√
√
√
√
I.e
√
. Define an invertible operator
and hence
)
and
(
√
)
are similar i.e.
.
Example 2.12.
Let
invertible
[
] and
[
] be two
matrix. Define
(square) matrices such that
∑
and
159
∑
where
is an
be the sum of diagonal entries
of matrices of
and
respectively. Then, since
and
are similar, they have the same characteristic
polynomial. In addition, they have the same trace i.e. trace of
matrix is similarity invariant.
Solution
Assume
.Then we have
[
I.e.
and
By
]
have got the same characteristic polynomial. Let
we have that
. So
multiplicities), say
and
and
be two similar
matrices.
have got the same eigenvalues (together with their
. Now since the trace of an
matrix coincides with the sum of its
∑
eigenvalues,
where
are the eigenvalues of
(counted with their multiplicities).
∑
implies that
as required.
Example 2.13.
Let
be an operator on an
represent
dimensional vector space. If two
with respect to different bases of
then
and
] and
[
matrices
and
are similar.
Solution
Assume that the two
} and {
bases {
[
matrices
for
∑
( )
and
for
Next, for each j choose scalars
( )
∑
( )
∑
and
such that
∑
and note that
[
, we get
∑
∑
with respect to
} respectively. That is
∑
invertible. Using
] represent the operator
∑
∑
=∑
∑
160
∑
∑
and from
.
we obtain
] is
Therefore , ∑
∑
. In matrix notation, this equivalent to
which establishes that matrices
and
or
are similar.
Remark 2.14. The Lemma below is vital in proving the result that if two operators
and
are similar,
then they have equal spectrum.
Lemma 2.15. [4]. Suppose that
Lemma 2.16. [4]. Suppose that
and
and
is invertible, then
.
are similar operators on a Hilbert space
, then
and
have the same
a) Spectrum
b) Point spectrum
c) Approximate point spectrum.
Example 2.17. Consider two similar operators
and
given in example 2.11 above. We now show
that they have the same spectrum.
The spectrum of
and
Thus
is the set of their respective eigenvalues since
i.e. *
Using the same approach, we find that
+
{
i.e.
and
It follows that
are finite.
{
}.
} as required.
Remark 2.18. [8, Theorem 7.1-3]: In view of example 2.1.11 above, all operators (matrices)
representing a given linear operator
on a finite dimensional normed space
bases have the same spectra.
161
relative to different
The following results help us investigate the relationship between similarity and unitary equivalence for
normal operators:
[
]
is a unitary operator for any self adjoint operator
[
Theorem 3.2.
Assume that
where
Proof: By proposition 3.1 above,
and
]
are normal and
, then
is a unitary operator
,
where S is any self-adjoint operator. The hypothesis
number
for any natural
, so that for any
̅
̅
Define a function
̅
̅
̅
Since
.
. This function can be expressed as
̅
̅
̅
and
̅
since
By the normality of
are both unitary operators by
∀
yields
and
by
.
are both self-adjoint operators,
so that
ensures that
. Consequently,
̅
̅
and
is analytic and bounded
is a constant by Lioville’s Theorem i.e.
and
̅
for any
, so that
holds. By differentiating both sides with respect to
we obtain the desired result
Remark 3.3. The hypothesis of theorem 3.2 does not imply that
self-adjoint and
(
(
)
are
(
) , then by simple calculation, it is true that
.
Theorem 3.4. ([11]). If
with
and
is normal for consider:
)
but
, even when
is invertible, then
an isometry (which is in fact unitary) and
decomposition
in which
and
has a unique polar decomposition
. If
commute with each other.
162
is normal, then
has a polar
We are now ready to show that similar normal operators are unitarily equivalent in the theorem that
follows:
Theorem 3.5. Suppose that
,
. If
Proof:
are normal
is the polar decomposition of
From
.
Hence
so that
is invertible and
, then
by
.
Theorem
B
2.2.2.
Hence
, since
This
yields
The Unilateral Shift
On Hilbert space
{
}
family {
an operator
is said to be a unilateral shift if there exists an infinite sequence
of non-zero pairwise orthogonal subspaces of
}
spans
) and
maps each
By the above definition,
isometry
and
hence
and
symmetrically onto
and
unitary.
Therefore,
for every
and its adjoint
such that
(i.e. the orthogonal
.
is an isometry, thus a surjective
and
are
unitarily
equivalent
. This constant dimension is the multiplicity of
are identified with infinite matrices
 0 U1 * 0
0 0 U *
2


0 U3 *

0


⊕
0 0
U
 1 0
 0 U2 0

U3


0







of transformations, where every entry below (above) the main block diagonal in the matrix
(Equivalently ,
for every
).
163
that
. More so







is unitary and the remaining entries are all null. From the above matrices, we see that
so
The canonical unilateral shift on
Let
.
be a Hilbert space and
Consider the operator
be unitary so that
for every
⊕
=⊕
Since the composition and direct sum of unitary transformations are again unitary, it follows that
is
unitary and
0 0
I 0

0 I 0

I


Thus
0







such that
is unitary equivalent to
If
(i.e, if
multiplicity 1 acting on
basis for
which is a unilateral shift of multiplicity
acting on
in the case of a complex space) then we get a canonical unilateral shift of
and is the operator on
that shifts the canonical orthonormal
.
Consequently, we prove the following results of unitary equivalence for unilateral shifts:
Proposition 3.6. Two unilateral shifts are unitarily equivalent if and only if they have the same
multiplicity.
Proof
Let
and
be unilateral shifts which are unitarily equivalent. Then they are unitarily equivalent to
the canonical unilateral shifts of multiplicity
Conversely, let
and
and
have the same multiplicity,
canonical unilateral shifts of multiplicity,
Proposition 3.7
. Hence
. Hence
and
have the same multiplicity.
. Then they are unitarily equivalent to the
are unitarily equivalent .
If an operator is unitarily equivalent to a unilateral shift, then it is a unilateral shift
itself of the same multiplicity.
Proof
Let
be a unilateral shift of multiplicity
acting on a Hilbert space
countably infinite orthogonal family of unitarily equivalent subspaces of
164
and let {
⊕
} be the underlying
so that
for all
.Take an operator
transformation
Since
acting on a Hilbert space
such that
is unitary, it follows that
⊕
set
so that
is an orthogonal family ( {
.
} is orthogonal and
because the range of an isometry is closed. Moreover, {
inner product) of subspaces of
i.e.
. For each
and suppose that there exists a unitary
preserves
} spans
. Furthermore, since
for each . Finally, note that
Thus
is an isometry and so any restriction of it (in particular
Thus
is a unilateral shift with the same multiplicity of
(
Let
and
∀
)
from
to
A quasi-affinity from
If
affinity
to
once
to
is a linear one-to-one and bicontinuous
We recall that two operators
similar if there exists an affinity
such that
and
are said to be
and as a result,
is a bounded linear operator which is one-to-one and has a dense range.
and
, we say tha
such that
is a quasi-affine transform of
The operators
and
and
165
if there exists a quasi-
are called quasi-similar if they are
quasi-affine transforms of each other, i.e. if there exists quasi-affinities
such that
).
.
be Hilbert spaces. An affinity from
transformation
is an isometry for every
and
Theorem 4.1.1. (i) If
and
is a quasi-affinity from
to
and
is a quasi-affinity from
to
where
are finite dimensional then
(a)
is a quasi-affinity from
(b)
to .
is a quasi-affinity from
Proof (a) since
to
.
is a quasi-affinity, then the operator
is intertwined to the operator
by . That is
.
In like manner, if
, then
From
is intertwined to
.Putting
in
.That is
to
. That is
that is
on the right hand side of
which implies that
is a quasi-affinity from
.
(b)
implies that
for all
and
linear one-to-one continuous transformation from
exists and as a result,
and so
. Since a quasi-affinity is a
onto a dense linear manifold in
. But the map
, the inverse
is essentially the map
as required.
Theorem 4.1.2. If
(a)
, that is
we get
. Applying
gives
by
is a quasi-affine transform of
and
is a quasi-affine transform of , then
is a quasi-affine transform of
(b)
is a quasi-affine transform of
Proof:
.
a quasi-affine transform of
such that
Similarly,
that there exists a quasi-affinity
, that is
implies that there exists a quasi-affinity
a quasi-affine transform of
such that
is a quasi-affine transform of
implies
By Theorem 4.1.1,
with quasi-affinity
where
. Part (b) follows
from Theorem 4.1.1 (b).
Proposition 4.1.3. If
from
to ) . Moreover
is a quasi-affinity from
to
then
√
is a quasi-affinity on
extends by continuity to a unitary transformation
166
from
to .
(i.e.
Lemma 4.1.4 [8, Lemma 2.6-11]. Let
and
and
be quasi-affinities where
are finite dimensional Hilbert spaces. Then the inverse
exists and
of the composite
.
Proposition 4.1.5. If a unitary operator
unitary operator
Proof:
on a Hilbert space
Let
on a Hilbert space
is the quasi-affine transform of a
then
and
are unitarily equivalent.
be
a
quasi-affinity.
implies that
From
Then
.
and
we obtain
and by iteration
for every polynomial
uniformly on the interval 0
and
} be a sequence of polynomials tending to
Let {
. Then
that we obtain a limit relation
From
hence
converges (in the operator norm) to
.
) it follows that
results that
so
because |
. By Proposition 4.1.3 above
is unitary and hence
is dense in
and
, it
are unitarily
equivalent.
Theorem 4.1.6. Quasi-similarity is an equivalence relation on the class of all operators.
Proof: Let
be a finite dimensional Hilbert space and let
Then
. First we show
and
where
invertible operators. Choosing
Now suppose that
are quasi-
(without loss of generality) we have that
. We show that
Thus
and
. Since
exists quasi-affinities
i.e.
and
such that
and
i.e.
. Hence
Suppose
and
. Then we show that
.
There exist quasi-invertible operators
and
respectively such that
and
and
and
.
167
Putting equation
in
we have
and
that is
and
It then follows from
where
that
and
i.e.
and
as required.
The following Theorem enables us obtain an example of quasi-similar operators:
Theorem 4.1.7 [7, Theorem 2.5]. Suppose that for each
and
and operators
in some index set
and
there are Hilbert spaces
respectively which are quasi-similar. Let
be the operator
∑
⊕
∑
⊕
acting on the Hilbert space which is the direct sum of the spaces
Example 4.1.8. Let
and
dimensional Hilbert space
similar. If
∑
where
∑
and
⊕
Then
is quasi-similar to
be unilateral shift operators with weights
. Then
.
and
is the Jordan canonical form for
∑
then by the above Theorem
and
respectively on an
and so
and
are
is quasi-similar to
.
Recall that an operator
intertwines
to
if
intertwined to , then there exists an operator with dense range intertwining
. If
to
is densely
.
The Lemma below gives a sufficient condition for transferring nontrivial invariant subspaces from
to
whenever
is densely intertwined to
.
Lemma 4.1.9 [9, Lemma 4.1]. Let
Suppose
,
and
is a nontrivial invariant subspace for
inverse image of
under
If
be such that
and
.
{ } the
is a nontrivial invariant subspace for
Remark 4.1.10. If the intertwining operator
is a nontrivial invariant subspace for
is surjective, (i.e.
) then
. Thus we have the following corollary:
Corollary 4.1.11 [9, Corollary 4.2]. If two operators are similar and if one of them has a non trivial
invariant subspace, then so has the other.
168
, { } is the set of all operators in
Definition 4.1.12. The commutator of
commutes with
that
, i.e.
{ }
{
}.
In other words, the commutant of an operator is the set of all operators intertwining it to itself.
{ } and
Claim:
Proof: { }
{ } whenever
{
{ } .
}. Now
, that is
and
that is
as required.
Actually { } is an operator algebra which contains the identity.
A linear manifold (subspace) of a normed space
is hyperinvariant for every
operator
in
on a normed space
{
{ }}
Hence
and for each vector
⊆
{ }
subspace, then
for every
(i.e.
Proposition 4.1.13. For each
Proof: Let
{ }}
such that
and
,there exists
every
(since
linear manifold of
that
{ }. Since
in
. Therefore, if
is a subspace of
⊆
{ }
has no nontrivial hyperinvariant
i.e.
if and only if
which is hyperinvariant for
Take an operator
.If
and consider the
{ }
then there exist operators
{ } If
{ } because
and hence
{ } such that
{ } , it follows that
is never empty unless
be an arbitrary vector in a normed space
{
set
set
. Then
{ } for every nonzero
{ } . For any
and so
for every scalar
and
{ } ). It follows that, as an algebra, { } is a linear space and hence
. Now take an arbitrary
(for
{ } . If
⊆
) .Thus
, then
and hence
for some
⊆
is a
{ } so
because
is
continuous.
Recalling that the closure of a linear manifold is a subspace and since
we conclude that ̅̅̅̅ is invariant for every
is an arbitrary operator in { }
{ } equivalently, ̅̅̅̅ is a hyperinvariant subspace for
169
Lemma 4.1.14.
Let
,
. Suppose
and
be such that
is a nontrivial hyperinvariant subspace for
{ } then
If
and
, ̅̅̅̅ is a nontrivial hyperinvariant
{ } and for each nonzero
subspace for .
Proof: By Proposition 4.1.13, { }
for every
{ } . If
{ } . We note that
for every
commutes with
so that
Now let
̅̅̅̅
commutes with
then
{ } for every
i.e.
be a nontrivial hyperinvariant subspace for . Since
hyperinvariant for
is
and take an arbitrary
so that
invariant, it follows that
must be in
then ̅̅̅̅̅̅̅̅̅̅
Therefore if ̅̅̅̅
for every
̅̅̅̅̅̅̅
ensure that there exists an
and as
because
̅̅̅̅̅̅̅̅̅
̅̅̅̅̅ ⊆
and
is closed and
is continuous.
and hence ̅̅̅̅̅̅̅̅̅̅
{ } ( i.e. if
such that if
implies
and since
⊆
), then
{ }. Hence ̅̅̅̅̅̅̅̅̅̅
{ }
is nonzero. Thus
{ } then ̅̅̅̅ is a nontrivial hyperinvariant for
and only if
for some
Thus in order to show that ̅̅̅̅ is a nontrivial hyperinvariant for , we must
in
{ }, Since
⊆
. There for
, it is
{ } . Since
. But ̅̅̅̅ is is a hyper invariant subspace for
for every
whenever
so that
for some
̅̅̅̅̅ ⊆ ̅
properly included in
{ }.
is hyperinvariant for
{ } . Take an arbitrary
whenever
{ }
for every nonzero
(for ̅̅̅̅
and
if
)
with ̅̅̅̅̅̅̅̅̅̅
and
such that ̅̅̅̅ is a nontrivial hyperinvariant subspace for
. Thus
Remark 4.1.15. In view of the above Lemma, if
then there exists
and
we have the following corollary:
Corollary 4.1.16. If two operators are quasi-similar and if one of them has a nontrivial hyperinvariant
subspace then so has the other .
170
[1] Clary S. Equality of spectra of quasisimilar hyponormal operators, Pro. Amer. Math. Soc. 53
(1975), No.1, 88-90.
[2] Douglas R.G. On the operator equation
(1969), 19-32.
and related topics, Acta Sci. Math (Szeged) 30
[3] Fialkol L.A. A note on quasisimilar operators, Acta Sci. Math. 39, 67-85.
[4] Halmos P.R. A Hilbert Space problem book, Van Nostrand, Princeston , N.J. , (1976).
[5] Herstain I.N. Topics in Algebra, University of Chicago, (1964).
[6] Hoover T.B. Hyperinvariant Subspaces for n-normal operators, Acta Sci. Math. , Vol.32 (1972)
pp109-119.
[7] Hoover T.B. Quasisimilarity of operators, Illinois. Math. 16 (1972), pp 678-688.
[8] Kryeyszig E. Introductory Functional Analysis with Applications, Wiley, New York (1978).
[9] Kubrusly C.S. An Introduction to Models and Decomposition in Operator Theory, Birkhauser (1997).
[10] Lambert A.J. Strictly Cyclic Operator Algebra, University of Michigan (1970).
[11] Nzimbi B.M., Pokhariyal G.P, and Khalagai J.M.
Operators, FJMS , Vol.28 No.2 (2008) ,pp 305-317.
A note on Similarity, and Equivalence of
[12] Stampfli J.G. Quasisimilarity of operators, Pro. Of the Royal Irish Academy. Section A:
Mathematical and Physical Sciences, Vol.81A, No.1 (1981) pp 109-119.
[13] Sz-Nagy B. and Foias C. Harmonic Analysis of operators on Hilbert Space, Translated from the
French and revised, North-Holland publishing co. ,Inc. ,New York; Akademiai Kiado, Budapest, 1970.
[14] Takayuki F. Invitation to Linear Operators, Taylor and Francis Inc, (2002).
[15] Williams L.R. Quasisimilarity and hyponormal operators, Jour.Oper. Th., 5 (1981), 127-702.
[16] Williams L.R. Equality of essential spectra of certain quasisimilar seminormal operators, Pro.
Amer. Math. Soc.78 (1980) 203-209.
171