Elements of Hilbert Spaces and Operator Theory

2017 ◽  
Author(s):  
Harkrishan Lal Vasudeva
Keyword(s):  
Hilbert Spaces ◽  
Operator Theory

scholarly journals More on Inequalities for Weaving Frames in Hilbert Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 141 ◽  
Author(s):  
Zhong-Qi Xiang
Keyword(s):  
Hilbert Spaces ◽  
Operator Theory ◽  
Triangle Inequality ◽  
Bounded Operator ◽  
Point Of View ◽  
Real Numbers ◽  
Linear Bounded Operator ◽  
Bessel Sequences ◽  
New Inequalities

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.

C∗-algebra consisting of all g-Bessel sequences in a Hilbert space

2017 ◽  
Vol 15 (01) ◽  
pp. 1750004
Author(s):  
Z. L. Chen ◽  
H. X. Cao ◽  
Z. H. Guo
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Small Perturbation ◽  
Hilbert Spaces ◽  
Operator Theory ◽  
Operator Space ◽  
Riesz Bases ◽  
C Algebra ◽  
Linear Bijection ◽  
Bessel Sequences

For Hilbert spaces [Formula: see text] and [Formula: see text], we use the notations [Formula: see text], [Formula: see text] and [Formula: see text] to denote the sets of all [Formula: see text]-Bessel sequences, [Formula: see text]-frames and Riesz bases in [Formula: see text] with respect to [Formula: see text], respectively. By defining a linear operation and a norm, we prove that [Formula: see text] becomes a Banach space and is isometrically isomorphic to the operator space [Formula: see text], where [Formula: see text]. In light of operator theory, it is proved that [Formula: see text] and [Formula: see text] are open sets in [Formula: see text]. This implies that both [Formula: see text]-frames and Riesz bases are stable under a small perturbation. By introducing a linear bijection [Formula: see text] from [Formula: see text] onto the [Formula: see text]-algebra [Formula: see text], a multiplication and an involution on the Banach space [Formula: see text] are defined so that [Formula: see text] becomes a unital [Formula: see text]-algebra that is isometrically isomorphic to the [Formula: see text]-algebra [Formula: see text], provided that [Formula: see text] and [Formula: see text] are isomorphic.

Block sequences and g-frames

2015 ◽  
Vol 13 (02) ◽  
pp. 1550009
Author(s):  
Renu Chugh ◽  
S. K. Sharma ◽  
Shashank Goel
Keyword(s):  
Hilbert Spaces ◽  
Operator Theory ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Block Sequence ◽  
Fusion Frames ◽  
Frame System ◽  
Necessary And Sufficient

Casazza and Kutyniok [Frames of subspaces, in Wavelets, Frames and Operator Theory, Contemporary Mathematics, Vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87–113] defined fusion frames in Hilbert spaces to split a large frame system into a set of (overlapping) much smaller systems and being able to process the data effectively locally within each sub-system. In this paper, we handle this problem using block sequences and generalized block sequences with respect to g-frames. Examples have been given to show their existence. A necessary and sufficient condition for a block sequence with respect to a g-frame to be a g-frame has been given. Finally, a sufficient condition for a generalized block sequence with respect to a g-frame to be a g-frame has been given.

scholarly journals On a Polyanalytic Approach to Noncommutative de Branges–Rovnyak Spaces and Schur Analysis

2021 ◽  
Vol 93 (4) ◽  
Author(s):  
Daniel Alpay ◽  
Fabrizio Colombo ◽  
Kamal Diki ◽  
Irene Sabadini
Keyword(s):  
Hardy Space ◽  
Hilbert Spaces ◽  
Operator Theory ◽  
Schur Multipliers ◽  
Hyperholomorphic Functions ◽  
Polyanalytic Functions ◽  
Points Of View ◽  
The Difference ◽  
Space Case ◽  
Class Of Functions

AbstractIn this paper we begin the study of Schur analysis and of de Branges–Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carathéodory multipliers.

scholarly journals Geometry and operator theory on quaternionic Hilbert spaces

2015 ◽  
Vol 6 (4) ◽  
pp. 226-246 ◽  
Author(s):  
Bingzhe Hou ◽  
Geng Tian
Keyword(s):  
Hilbert Spaces ◽  
Operator Theory
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