scholarly journals The invariant subspaces of S ⊕ S*

2020 ◽  
Vol 7 (1) ◽  
pp. 116-123
Author(s):  
Dan Timotin
Keyword(s):  
Hilbert Space ◽  
Invariant Subspaces ◽  
Unilateral Shift

AbstractUsing the tools of Sz.-Nagy–Foias theory of contractions, we describe in detail the invariant subspaces of the operator S ⊕ S*, where S is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.

scholarly journals Biquasitriangularity and spectral continuity

1985 ◽  
Vol 26 (2) ◽  
pp. 177-180 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Invariant Subspaces ◽  
Extension Property ◽  
Point Of View ◽  
Single Valued Extension Property ◽  
Continuity Theorem ◽  
Spectral Continuity ◽  
Necessary And Sufficient ◽  
Continuity Of The Spectrum

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

On Regular Generalized Sampling in T-Invariant Subspaces of a Hilbert Space: An Overview

2019 ◽  
Vol 41 (6) ◽  
pp. 685-709
Author(s):  
Antonio G. García ◽  
María J. Muñoz-Bouzo ◽  
Gerardo Pérez-Villalón
Keyword(s):  
Hilbert Space ◽  
Invariant Subspaces ◽  
Generalized Sampling

On Operator Algebras and Invariant Subspaces

1969 ◽  
Vol 21 ◽  
pp. 1178-1181 ◽  
Author(s):  
Chandler Davis ◽  
Heydar Radjavi ◽  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Elementary Proof ◽  
Operator Algebras ◽  
Equivalent Form ◽  
Invariant Subspaces ◽  
Complex Hilbert Space ◽  
Closed Algebra ◽  
Weakly Closed ◽  
Special Case

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.

Involutory *-antiautomorphisms in Toeplitz algebras

1988 ◽  
Vol 103 (3) ◽  
pp. 473-480
Author(s):  
P. J. Stacey
Keyword(s):  
Hilbert Space ◽  
Orthonormal Basis ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Unilateral Shift ◽  
Integral Power

Let H be a separable complex Hilbert space with orthonormal basis {ei: i ∈ ℕ}, let s be the unilateral shift defined by sei = ei+1 for each i and let K be the algebra of compact operators on H. The present paper classifies the involutory *-anti-automorphisms in the C*-algebra C*(sn, K) generated by K and a positive integral power sn of s. It is shown that, up to conjugacy by *-automorphisms, there are two such involutory *-antiautomorphisms when n is even and one when n is odd.

On Reflexivity of Algebras

1981 ◽  
Vol 33 (6) ◽  
pp. 1291-1308 ◽  
Author(s):  
Mehdi Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Natural Number ◽  
Separable Hilbert Space ◽  
Invariant Subspaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Direct Sum ◽  
Weakly Closed

For each natural number n we define to be the class of all weakly closed algebras of (bounded linear) operators on a separable Hilbert space H such that the lattice of invariant subspaces of and (alg lat )(n) are the same. (If A is an operator, A(n) denotes the direct sum of n copies of A; if is a collection of operators,. Also, alg lat denotes the algebra of all operators leaving all invariant subspaces of invariant.) In the first section we show that . In Section 2 we prove that every weakly closed algebra containing a maximal abelian self adjoint algebra (m.a.s.a.) is , and that . It is also shown that certain algebras containing a m.a.s.a. are necessarily reflexive.

Compact Operators in Reductive Algebras

1975 ◽  
Vol 27 (1) ◽  
pp. 152-154
Author(s):  
Edward A. Azoff
Keyword(s):  
Hilbert Space ◽  
Invariant Subspaces ◽  
Compact Operators ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Term Algebra

Let be a Hilbert space and denote the collection of (bounded, linear) operators on by . Throughout this paper, the term ‘algebra’ will refer to a subalgebra of ; unless otherwise stated, it will not be assumed to contain I or to be closed in any topology.An algebra is said to be transitive if it has no non-trivial invariant subspaces. The following lemma has revolutionized the study of transitive algebras. For a pr∞f and a general discussion of its implications, the reader is referred to [5].

A proof of Hartman's theorem on compact Hankel operators

1975 ◽  
Vol 78 (3) ◽  
pp. 447-450 ◽  
Author(s):  
F. F. Bonsall ◽  
S. C. Power
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Hankel Operator ◽  
Hankel Operators ◽  
Unilateral Shift ◽  
Space Model ◽  
Model Free ◽  
Multiplicity One ◽  
Image Position ◽  
Algebra Of Operators

Let U be a unilateral shift of arbitrary (perhaps uncountable) multiplicity on a Hilbert space. Following Rosenblum (5), an operator A is said to be a Hankel operator relative to U ifHartman (2) has characterized the compact Hankel operators relative to the unilateral shift of multiplicity one as the Hankel operators with symbol in H∞ + C(T). Using the usual function space model for representing the unilateral shift, Page ((4), theorem 10) has extended Hartman's theorem to unilateral shifts of countable multiplicity. We give a model-free proof of Hartman's theorem which applies to shifts of arbitrary multiplicity. The proof turns on the observation that a compact operator acts compactly on a certain algebra of operators.

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