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.

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.

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.

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).

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 .

Unitarily Invariant Operator Norms

1983 ◽  
Vol 35 (2) ◽  
pp. 274-299 ◽  
Author(s):  
C.-K. Fong ◽  
J. A. R. Holbrook
Keyword(s):  
Hilbert Space ◽  
Convex Functions ◽  
Invariant Operator ◽  
Space Operator ◽  
Numerical Radius ◽  
The Past ◽  
Operator Norms ◽  
Power Inequality ◽  
The Many ◽  
Image Position

1.1. Over the past 15 years there has grown up quite an extensive theory of operator norms related to the numerical radius1of a Hilbert space operator T. Among the many interesting developments, we may mention:(a) C. Berger's proof of the “power inequality”2(b) R. Bouldin's result that3for any isometry V commuting with T;(c) the unification by B. Sz.-Nagy and C. Foias, in their theory of ρ-dilations, of the Berger dilation for T with w(T) ≤ 1 and the earlier theory of strong unitary dilations (Nagy-dilations) for norm contractions;(d) the result by T. Ando and K. Nishio that the operator radii wρ(T) corresponding to the ρ-dilations of (c) are log-convex functions of ρ.

scholarly journals On the Non-Hypercyclicity of Normal Operators, Their Exponentials, and Symmetric Operators

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 903
Author(s):  
Marat V. Markin ◽  
Edward S. Sichel
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Normal Operators ◽  
Symmetric Operators ◽  
Straightforward Proof

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator A in a complex Hilbert space as well as of the collection e t A t ≥ 0 of its exponentials, which, under a certain condition on the spectrum of A, coincides with the C 0 -semigroup generated by it. We also establish non-hypercyclicity for symmetric operators.

scholarly journals States which have a trace-like property relative to a C*-subalgebra of B(H)

1976 ◽  
Vol 17 (2) ◽  
pp. 158-160
Author(s):  
Guyan Robertson
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Operator Theory ◽  
Linear Operators ◽  
Operator Norm ◽  
Identity Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.

Spectral Inclusion and C.N.E.

1982 ◽  
Vol 34 (4) ◽  
pp. 883-887 ◽  
Author(s):  
A. R. Lubin
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Normal Extension ◽  
Unitary Equivalence ◽  
Bounded Linear ◽  
Multi Index ◽  
Reducing Subspace ◽  
Normal Operators ◽  
Image Position ◽  
Minimal Normal Extension

1. An n-tuple S = (S1, …, Sn) of commuting bounded linear operators on a Hilbert space H is said to have commuting normal extension if and only if there exists an n-tuple N = (N1, …, Nn) of commuting normal operators on some larger Hilbert space K ⊃ H with the restrictions Ni|H = Si, i = 1, …, n. If we takethe minimal reducing subspace of N containing H, then N is unique up to unitary equivalence and is called the c.n.e. of S. (Here J denotes the multi-index (j1, …, jn) of nonnegative integers and N*J = N1*jl … Nn*jn and we emphasize that c.n.e. denotes minimal commuting normal extension.) If n = 1, then S1 = S is called subnormal and N1 = N its minimal normal extension (m.n.e.).

scholarly journals Inequality of Taikov type for powers of normal operators in Hilbert space

2021 ◽  
Vol 19 ◽  
pp. 3
Author(s):  
V.F. Babenko ◽  
R.O. Bilichenko
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Intermediate Derivative ◽  
Higher Derivative ◽  
Normal Operators

The Taikov inequality, which estimates $L_{\infty}$-norm of intermediate derivative by $L_2$-norms of a function and its higher derivative, is extended on arbitrary powers of normal operator acting in Hilbert space.

scholarly journals On (P*-N) quasi normal operators Of order "n" In Hilbert space

2021 ◽  
Vol 32 (1) ◽  
pp. 10
Author(s):  
Salim Dawood M. ◽  
Jaafer Hmood Eidi
Keyword(s):  
Functional Analysis ◽  
Hilbert Space ◽  
Normal Operator ◽  
Sufficient Conditions ◽  
Basic Operation ◽  
Properties And Characterization ◽  
Normal Operators

Through this paper, we submitted  some types of quasi normal operator is called be (k*-N)- quasi normal operator of order n defined on a Hilbert space H, this concept is generalized of some kinds of  quasi normal operator appear recently form most researchers in the  field of functional analysis, with some properties  and characterization of this operator   as well as, some basic operation such as addition and multiplication of these operators had been given, finally the relationships of this operator proved with some examples to illustrate conversely and introduce the sufficient conditions to satisfied this case with other types had been studied.

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