scholarly journals On Pompeiu Cebysev Type Inequalities for Positive Linear Maps of Selfadjoint Operators in Inner Product Spaces

2018 ◽  
Vol 15 ◽  
pp. 8081-8092
Author(s):  
Mohammad W Alomari
Keyword(s):  
Hilbert Spaces ◽  
Inner Product ◽  
Product Spaces ◽  
Selfadjoint Operators ◽  
Inner Product Spaces ◽  
Linear Maps ◽  
Positive Linear Maps

In this work, generalizations of some inequalities for continuous synchronous (h-asynchronous) functions of linear bounded selfadjoint operators under positive linear maps in Hilbert spaces are proved.

scholarly journals Some inequalities for the norm and the numerical radius of linear operators in Hilbert spaces

2008 ◽  
Vol 39 (1) ◽  
pp. 1-7 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Operator Norm ◽  
Inner Product ◽  
Product Spaces ◽  
Numerical Radius ◽  
Inner Product Spaces

In this paper various inequalities between the operator norm and its numerical radius are provided. For this purpose, we employ some classical inequalities for vectors in inner product spaces due to Buzano, Goldstein-Ryff-Clarke, Dragomir-S ´andor and the author.

scholarly journals Operator Popoviciu’s inequality for superquadratic and convex functions of selfadjoint operators in Hilbert spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 313-324
Author(s):  
Mohammad W. Alomari
Keyword(s):  
Hilbert Spaces ◽  
Convex Functions ◽  
Positive Operators ◽  
Selfadjoint Operators ◽  
Popoviciu’S Inequality ◽  
Linear Maps ◽  
Superquadratic Functions ◽  
Operator Version ◽  
Positive Linear Maps

AbstractIn this work, an operator version of Popoviciu’s inequality for positive operators on Hilbert spaces under positive linear maps for superquadratic functions is proved. Analogously, using the same technique, an operator version of Popoviciu’s inequality for convex functions is obtained. Some other related inequalities are also presented.

Topologies on Generalized Inner Product Spaces

1969 ◽  
Vol 21 ◽  
pp. 158-169 ◽  
Author(s):  
Eduard Prugovečki
Keyword(s):  
Hilbert Spaces ◽  
Quantum Physics ◽  
Point Of View ◽  
Inner Product ◽  
Product Spaces ◽  
Topological Vector Spaces ◽  
Fundamental Properties ◽  
Inner Product Spaces ◽  
Algebraic Generalization ◽  
Dirac's Formalism

In the present note we introduce a straightforward algebraic generalization of inner product spaces, which we appropriately name generalized inner product (GIP) spaces. In the same fashion in which different topologies :an be introduced in inner product spaces, adequate topologies can be introduced in GIP spaces in such a manner that topological vector spaces are obtained. We enumerate and derive some fundamental properties of different topologies in GIP spaces, having primarily in mind their possible later application to quantum physics.The desirability of having in quantum physics more general structures than Hilbert spaces (in which quantum mechanics is usually formulated) is suggested by Dirac's formalism (2), which deals with “unnormalizable” vectors. Unfortunately, although this formalism is very elegant from the point of view of the facili ty of dealing with its symbolism, it completely lacks in mathematical rigour.

scholarly journals Moore-Penrose inverse of some linear maps on infinite-dimensional vector spaces

2020 ◽  
Vol 36 (36) ◽  
pp. 570-586 ◽  
Author(s):  
Fernando Pablos Romo ◽  
Víctor Cabezas Sánchez
Keyword(s):  
Linear System ◽  
Vector Spaces ◽  
Inner Product ◽  
Dimensional Vector ◽  
Product Spaces ◽  
Infinite Dimensional ◽  
Inner Product Spaces ◽  
Linear Maps ◽  
Infinite Linear System ◽  
Infinite Linear

The aim of this work is to characterize linear maps of infinite-dimensional inner product spaces where the Moore-Penrose inverse exists. This MP inverse generalizes the well-known Moore-Penrose inverse of a matrix $A\in \text{Mat}_{n\times m} ({\mathbb C})$. Moreover, a method for the computation of the MP inverse of some endomorphisms on infinite-dimensional vector spaces is given. As an application, the least norm solution of an infinite linear system from the Moore-Penrose inverse offered is studied.

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