(k, m, n)-PARTIALLY ISOMETRIC OPERATORS: A NEW GENERALIZATION OF PARTIAL ISOMETRIES
Abstract
A great deal of elegant work has been done for partial isometries. They form
an attractive class of operators which arise in many applications, in particular they enter in
quantum physics [7, 27]. The quantization we talked about is a long-established physicist
concept that is analogous to the Jordan-Schwinger map. Instead of creation-annihilation
operators, a model for this quantization employed the countable family of orthogonal partial isometries of separable Hilbert space as its building blocks [6]. They enter also in pure
mathematics for which they furnish an in-depth studied extension of isometries. Thanks
to early work of Erd´elyi [13, 14], Halmos and McLaughlin [23, 24], among others; they
have played a fundamental role in operator theory: especially in the theory of the polar
decomposition of operators, and in the dimension theory of Von Neumann algebras. In
the present article, we introduce a new class of operators which will be called the class of
(k, m, n)-partial isometries that constitutes a generalization of the concept of partial isometries. We discuss the most interesting results concerning this class by extending some known
results obtained for partial isometries and exploring new results on partial isometries. We
study the connection of this new class of operators with classical notions of operators, such
partial isometries, power partial isometries, paranormal, semi-regular and quasi-Fredholm.
We investigate some basic properties and structure theorems. Moreover, we give spectral
theorems for this class of operators.
an attractive class of operators which arise in many applications, in particular they enter in
quantum physics [7, 27]. The quantization we talked about is a long-established physicist
concept that is analogous to the Jordan-Schwinger map. Instead of creation-annihilation
operators, a model for this quantization employed the countable family of orthogonal partial isometries of separable Hilbert space as its building blocks [6]. They enter also in pure
mathematics for which they furnish an in-depth studied extension of isometries. Thanks
to early work of Erd´elyi [13, 14], Halmos and McLaughlin [23, 24], among others; they
have played a fundamental role in operator theory: especially in the theory of the polar
decomposition of operators, and in the dimension theory of Von Neumann algebras. In
the present article, we introduce a new class of operators which will be called the class of
(k, m, n)-partial isometries that constitutes a generalization of the concept of partial isometries. We discuss the most interesting results concerning this class by extending some known
results obtained for partial isometries and exploring new results on partial isometries. We
study the connection of this new class of operators with classical notions of operators, such
partial isometries, power partial isometries, paranormal, semi-regular and quasi-Fredholm.
We investigate some basic properties and structure theorems. Moreover, we give spectral
theorems for this class of operators.
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