scholarly journals Relatively Orthocomplemented Skew Nearlattices in Rickart Rings

2015 ◽  
Vol 48 (4) ◽  
Author(s):  
Jānis Cīırulis
Keyword(s):  
Hilbert Space ◽  
Upper Bound ◽  
Initial Segment ◽  
Binary Operation ◽  
Normal Band ◽  
Natural Order ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Star Order ◽  
The One

AbstractA class of (right) Rickart rings, called strong, is isolated. In particular, every Rickart *-ring is strong. It is shown in the paper that every strong Rickart ring R admits a binary operation which turns R into a right normal band having an upper bound property with respect to its natural order ≤; such bands are known as right normal skew nearlattices. The poset (R, ≤) is relatively orthocomplemented; in particular, every initial segment in it is orthomodular.The order ≤ is actually a version of the so called right-star order. The one-sided star orders are well-investigated for matrices and recently have been generalized to bounded linear Hilbert space operators and to abstract Rickart *-rings. The paper demonstrates that they can successfully be treated also in Rickart rings without involution.

scholarly journals Bounded point evaluations for cyclic Hilbert space operators

2003 ◽  
Vol 4 (2) ◽  
pp. 301
Author(s):  
A. Bourhim
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Point Evaluations ◽  
Analytic Bounded Point Evaluations

<p>In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams.</p>

scholarly journals Two conditions for subnormality of unbounded operators

2001 ◽  
Vol 43 (1) ◽  
pp. 23-28
Author(s):  
Jan Niechwiej
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
The Real ◽  
Completely Monotonic ◽  
Binomial Expansion ◽  
Hilbert Space Operators ◽  
The One

We give two new sufficient conditions for unbounded Hilbert space operators to be subnormal. The first assumes that the sequence //Tnf//2 on a suitable subset of the domain is completely monotonic, the second is similar to the one given by Lambert in [3] for bounded operators and involves the sequence of binomial expansion of the real part of the operator.

scholarly journals A Note on Real Operator Monotone Functions

2020 ◽  
Author(s):  
Marcell Gaál ◽  
Miklós Pálfia
Keyword(s):  
Hilbert Space ◽  
Functional Calculus ◽  
Monotone Functions ◽  
Open Convex ◽  
Completely Positive ◽  
Bounded Linear ◽  
Operator Monotone ◽  
Hilbert Space Operators ◽  
Operator Monotone Functions

Abstract In this paper, we initiate the study of real operator monotonicity for functions of tuples of operators, which are multivariate structured maps with a functional calculus called free functions that preserve the order between real parts (or Hermitian parts) of bounded linear Hilbert space operators. We completely characterize such functions on open convex free domains in terms of ordinary operator monotone free functions on self-adjoint domains. Further assuming the more stringent free holomorphicity, we prove that all such functions are affine linear with completely positive nonconstant part. This problem has been proposed by David Blecher at the biannual OTOA conference held in Bangalore in December 2016.

scholarly journals Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov

2020 ◽  
Vol 7 (1) ◽  
pp. 133-154
Author(s):  
V. Müller ◽  
Yu. Tomilov
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Unit Vector ◽  
Linear Operators ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Single Operator ◽  
Orbits Of Operators

AbstractWe present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vector x ∈ H such that |〈Tjx, x〉| is “large” for all j = 1, . . . , n.

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 About the Cauchy–Bunyakovsky–Schwarz Inequality for Hilbert Space Operators

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 305
Author(s):  
Nicuşor Minculete
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Schwarz Inequality ◽  
Linear Operators ◽  
Real Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Bohr’S Inequality ◽  
Symmetric Shape

The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators.

scholarly journals Additive perturbations and multiplicative perturbations for the core inverse of bounded linear operator in Hilbert space

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6131-6144
Author(s):  
Fapeng Du ◽  
Zuhair Nashed
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Upper Bound ◽  
Bounded Linear Operator ◽  
Upper Bounds ◽  
Bounded Linear ◽  
The Core ◽  
Core Inverse

In this paper, we present some characteristics and expressions of the core inverse A# of bounded linear operator A in Hilbert spaces. Additive perturbations of core inverse are investigated under the condition R( ?)?N(A#) = {0} and an upper bound of ||?#-A#|| is obtained. We also discuss the multiplicative perturbations. The expressions of core inverse of perturbed operator T = EAF and the upper bounds of ||T#-A#|| are obtained too.

IMPROVED INEQUALITIES FOR THE NUMERICAL RADIUS: WHEN INVERSE COMMUTES WITH THE NORM

2018 ◽  
Vol 97 (2) ◽  
pp. 293-296 ◽  
Author(s):  
BRYAN E. CAIN
Keyword(s):  
Hilbert Space ◽  
Numerical Radius ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
New Inequalities

New inequalities relating the norm $n(X)$ and the numerical radius $w(X)$ of invertible bounded linear Hilbert space operators were announced by Hosseini and Omidvar [‘Some inequalities for the numerical radius for Hilbert space operators’, Bull. Aust. Math. Soc.94 (2016), 489–496]. For example, they asserted that $w(AB)\leq$$2w(A)w(B)$ for invertible bounded linear Hilbert space operators $A$ and $B$. We identify implicit hypotheses used in their discovery. The inequalities and their proofs can be made good by adding the extra hypotheses which take the form $n(X^{-1})=n(X)^{-1}$. We give counterexamples in the absence of such additional hypotheses. Finally, we show that these hypotheses yield even stronger conclusions, for example, $w(AB)=w(A)w(B)$.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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