scholarly journals A structure theorem for operators with closed range

1978 ◽  
Vol 18 (2) ◽  
pp. 169-186
Author(s):  
James Guyker
Keyword(s):  
Hilbert Space ◽  
Structure Theorem ◽  
Linear Transformations ◽  
Partial Isometries ◽  
Closed Range

A characterization has previously been given for linear transformations in Hilbert space whose first N + 1 powers are partial isometries. An analogous characterization is now obtained for transformations whose first N+ 1 powers have closed ranges. A hypothesis (that transformations have no isometric part) is found to be unnecessary in previous work.

scholarly journals Maximal nonnegative and $\theta$-accretive extensions of a positive definite linear relation

2020 ◽  
Vol 12 (2) ◽  
pp. 289-296
Author(s):  
O.G. Storozh
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Linear Relation ◽  
Differential Operators ◽  
Boundary Value ◽  
Positive Definite ◽  
Complex Hilbert Space ◽  
Multivalued Operator ◽  
Linear Transformations ◽  
Accretive Extension

Let $L_{0}$ be a closed linear positive definite relation ("multivalued operator") in a complex Hilbert space. Using the methods of the extension theory of linear transformations in a Hilbert space, in the terms of so called boundary value spaces (boundary triplets), i.e. in the form that in the case of differential operators leads immediately to boundary conditions, the general forms of a maximal nonnegative, and of a proper maximal $\theta$-accretive extension of the initial relation $L_{0}$ are established.

Similarity Classes of Linear Transformations

2020 ◽  
Vol 87 (3-4) ◽  
pp. 148
Author(s):  
Puja Bharti ◽  
Jagmohan Tanti
Keyword(s):  
Scalar Field ◽  
Vector Space ◽  
Characteristic Polynomial ◽  
Principal Ideal ◽  
Structure Theorem ◽  
Principal Ideal Domain ◽  
Linear Transformations ◽  
Ideal Domain ◽  
Number Of Classes ◽  
Finitely Generated Modules

In this paper, we investigate the similarity classes of linear transformations on a vector space using structure theorem for finitely generated modules over a principal ideal domain. We also establish formulae to count similarity classes with a given polynomial as a characteristic polynomial and to count total number of classes when the scalar field is finite.

Products of idempotent linear transformations

1985 ◽  
Vol 100 (1-2) ◽  
pp. 123-138 ◽  
Author(s):  
M. A. Reynolds ◽  
R. P. Sullivan
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bounded Operators ◽  
Linear Transformations ◽  
Arbitrary Vector ◽  
Dimensional Vector Space ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional

SynopsisIn 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

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