Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

Some results on generalized finite operators and range kernel orthogonality in Hilbert spaces

2021 ◽  
Vol 54 (1) ◽  
pp. 318-325
Author(s):  
Nadia Mesbah ◽  
Hadia Messaoudene ◽  
Asma Alharbi
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Generalized Derivation ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Similarity Orbit

Abstract Let ℋ {\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ ( ℋ ) {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ {\mathcal{ {\mathcal H} }} . In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators ( A , B ) ∈ ℬ ( ℋ ) × ℬ ( ℋ ) \left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥ A X − X B − I ∥ ≥ 1 , for all X ∈ ℬ ( ℋ ) . \parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

Operators of the form PAQ-QAP

1968 ◽  
Vol 20 ◽  
pp. 1353-1361 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

In this note the Hilbert spaces under consideration are complex, and the operators referred to are bounded, linear operators. If is a Hilbert space, then the algebra of all operators on is denoted by .It is known (1) that if is any Hilbert space, then the class of commutators on , i.e., the class of all operators that can be written in the form PQ — QP for some , can be exactly described. A similar problem is that of characterizing all operators on that can be written in the form for some .

scholarly journals Some vector inequalities for two operators in Hilbert spaces with applications

2016 ◽  
Vol 8 (1) ◽  
pp. 75-92
Author(s):  
Sever S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Nonnegative Operator

AbstractIn this paper we establish some vector inequalities for two operators related to Schwarz and Buzano results. We show amongst others that in a Hilbert space H we have the inequality $${1 \over 2}\left[ {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{x}},{\rm{x}}} \right\rangle ^{1/2} \left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over 2}{\rm{y}},{\rm{y}}} \right\rangle ^{1/2} + \left| {\left\langle {{{\left| {\rm{A}} \right|^2 + \left| {\rm{B}} \right|^2 } \over {\rm{2}}}} {\rm{x}},{\rm{y}}\right\rangle } \right|} \right] \ge \left| {\left\langle {{\mathop{\rm Re}\nolimits} ({\rm{B}}*{\rm{A}})\,{\rm{x}},{\rm{y}}} \right\rangle } \right|$$ for A, B two bounded linear operators on H such that Re (B*A) is a nonnegative operator and any vectors x, y ∈ H.Applications for norm and numerical radius inequalities are given as well.

Commutators of Operators on Hilbert Space

1965 ◽  
Vol 17 ◽  
pp. 695-708 ◽  
Author(s):  
Arlen Brown ◽  
P. R. Halmos ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Bilateral Shift

The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.

scholarly journals More on -Normal Operators in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Rasoul Eskandari ◽  
Farzollah Mirzapour ◽  
Ali Morassaei
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Operator Norm ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Normal Operators

We study some properties of -normal operators and we present various inequalities between the operator norm and the numerical radius of -normal operators on Banach algebraℬ() of all bounded linear operators , where is Hilbert space.

scholarly journals A characterization of real and complex Hilbert spaces among all normed spaces

1983 ◽  
Vol 27 (3) ◽  
pp. 339-345
Author(s):  
J. Vukman
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Identity Operator ◽  
Normed Spaces ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Dimensional Range

Let X be a real or complex normed space and L(X) the algebra of all bounded linear operators on X. Suppose there exists a *-algebra B(X) ⊂ L(X) which contains the identity operator I and all bounded linear operators with finite-dimensional range. The main result is: if each operator U ∈ B(X) with the property U*U = UU* = I has norm one then X is a Hilbert space.

Sign in / Sign up

Export Citation Format

Share Document