scholarly journals A Joint spectral theorem for unbounded normal opertors

1983 ◽  
Vol 34 (2) ◽  
pp. 203-213 ◽  
Author(s):  
A. B. Patel
Keyword(s):  
Hilbert Space ◽  
Spectral Theorem ◽  
Normal Operators

AbstractA joint spectral theorem for an n-tuple of doubly commuting unbounded normal operators in a Hilbert space is proved by using the techniques of GB*-algebras.

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.

On the spectral theorem for normal operators

1970 ◽  
Vol 68 (2) ◽  
pp. 393-400 ◽  
Author(s):  
R. G. Douglas ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Measure Theory ◽  
Normal Operator ◽  
The Other ◽  
Spectral Theorem ◽  
Ideal Space ◽  
Von Neumann ◽  
Normal Operators ◽  
Abelian Von Neumann Algebra

It has been known for some time that one can construct a proof of the spectral theorem for a normal operator on a Hilbert space by applying the Gelfand representation theorem to the Abelian von Neumann algebra generated by the normal operator, and using the fact that the maximal ideal space of an Abelian von Neumann algebra is extremely disconnected. This, in fact, is the spirit of the monograph (8). On the other hand, it is difficult to find in print accounts of the spectral theorem from this viewpoint and, in particular, the treatment in (8) uses a considerable amount of measure theory and does not have the proof of the spectral theorem as its main objective.

SUPPORT OF A JOINT RESOLUTION OF IDENTITY AND THE PROJECTION SPECTRAL THEOREM

2003 ◽  
Vol 06 (04) ◽  
pp. 549-561 ◽  
Author(s):  
ARTEM D. PULEMYOTOV
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Spectral Expansion ◽  
Full Measure ◽  
Spectral Theorem ◽  
Outer Measure ◽  
Normal Operators ◽  
Joint Resolution

Let A = (Ax)x ∈ Xbe a family of commuting normal operators in a separable Hilbert space H0. Obtaining the spectral expansion of A involves constructing of the corresponding joint resolution of identity E. The support supp E is not, in general, a set of full measure. This causes numerous difficulties, in particular, when proving the projection spectral theorem, i.e. the main theorem about the expansion in generalized joint eigenvectors. In this work, we show that supp E has a full outer measure under the conditions of the projection spectral theorem. Using this result, we simplify the proof of the theorem and refine its assertions.

scholarly journals A weaker condition for normality

1994 ◽  
Vol 36 (2) ◽  
pp. 249-253
Author(s):  
Ian Doust
Keyword(s):  
Hilbert Space ◽  
Orthogonal Projection ◽  
Operator Theory ◽  
Normal Operator ◽  
Borel Subset ◽  
Spectral Theorem ◽  
Space Operator ◽  
Countable Additivity ◽  
Normal Operators ◽  
Image Position

One of the most important results of operator theory is the spectral theorem for normal operators. This states that a normal operator (that is, a Hilbert space operator T such that T*T= TT*), can be represented as an integral with respect to a countably additive spectral measure,Here E is a measure that associates an orthogonal projection with each Borel subset of ℂ. The countable additivity of this measure means that if x Eℋ can be written as a sum of eigenvectors then this sum must converge unconditionally.

Formally Normal Operators Having no Normal Extensions

1965 ◽  
Vol 17 ◽  
pp. 1030-1040 ◽  
Author(s):  
Earl A. Coddington
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Null Space ◽  
Normal Operator ◽  
Closed Linear Operator ◽  
Normal Operators ◽  
The Given ◽  
Densely Defined ◽  
Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

scholarly journals Some inequalities for trace class operators via a Kato’s result

2017 ◽  
Vol 11 (01) ◽  
pp. 1850004
Author(s):  
S. S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Trace Class Operators ◽  
Kato’S Inequality ◽  
New Inequalities

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

Sign in / Sign up

Export Citation Format

Share Document