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.

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.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

scholarly journals States which have a trace-like property relative to a C*-subalgebra of B(H)

1976 ◽  
Vol 17 (2) ◽  
pp. 158-160
Author(s):  
Guyan Robertson
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Operator Theory ◽  
Linear Operators ◽  
Operator Norm ◽  
Identity Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.

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.

scholarly journals A characterization of spectral operators on Hilbert spaces

1982 ◽  
Vol 23 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Ernst Albrecht
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Spectral Operator ◽  
Bounded Linear ◽  
Image Position

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following propertyConversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

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 ORTHOGONALITY AND PARALLELISM OF OPERATORS ON VARIOUS BANACH SPACES

2018 ◽  
Vol 106 (2) ◽  
pp. 160-183 ◽  
Author(s):  
T. BOTTAZZI ◽  
C. CONDE ◽  
M. S. MOSLEHIAN ◽  
P. WÓJCIK ◽  
A. ZAMANI
Keyword(s):  
Banach Spaces ◽  
Compact Operators ◽  
Linear Operators ◽  
Identity Operator ◽  
Uniformly Convex ◽  
Bounded Linear Operators ◽  
Norm Inequalities ◽  
Bounded Linear ◽  
Uniformly Convex Spaces

We present some properties of orthogonality and relate them with support disjoint and norm inequalities in $p$-Schatten ideals. In addition, we investigate the problem of characterization of norm-parallelism for bounded linear operators. We consider the characterization of the norm-parallelism problem in $p$-Schatten ideals and locally uniformly convex spaces. Later on, we study the case when an operator is norm-parallel to the identity operator. Finally, we give some equivalence assertions about the norm-parallelism of compact operators. Some applications and generalizations are discussed for certain operators.

scholarly journals Exact and Approximate Operator Parallelism

2015 ◽  
Vol 58 (1) ◽  
pp. 207-224 ◽  
Author(s):  
Ali Zamani ◽  
Mohammad Sal Moslehian
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Operator Norm ◽  
Normed Spaces ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
C Algebra ◽  
Hilbert Space Operators ◽  
The Relationship

AbstractExtending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra B(ℋ) of bounded linear operators acting on a Hilbert space H . Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on B(ℋ). We also characterize the parallel elements of a C*-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert C*-module.

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.

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