Lifting the Commutant of a Subnormal Operator

1979 ◽  
Vol 31 (1) ◽  
pp. 148-156 ◽  
Author(s):  
Robert F. Olin ◽  
James E. Thomson
Keyword(s):  
Hilbert Space ◽  
Subnormal Operator ◽  
Basic Material ◽  
Normal Extension ◽  
Subnormal Operators ◽  
Double Commutant ◽  
Minimal Normal Extension

Let S be a subnormal operator on a Hilbert space ℋ and let N be its minimal normal extension on the Hilbert space ℋ. (We refer the reader to [5, 15] for the basic material on subnormal operators.) Denote the commutant and double commutant of an operator T by ﹛T﹜’ and ﹛T﹜”, respectively.

scholarly journals The spectrum of orthogonal sums of subnormal pairs

1988 ◽  
Vol 30 (1) ◽  
pp. 11-15 ◽  
Author(s):  
K. Rudol
Keyword(s):  
Invariant Subspace ◽  
Spectral Theory ◽  
Existence And Uniqueness ◽  
Normal Extension ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Basic Properties ◽  
Subspace Theorem ◽  
Subnormal Operators ◽  
Minimal Normal Extension

This note provides yet another example of the difficulties that arise when one wants to extend the spectral theory of subnormal operators to subnormal tuples. Several basic properties of a subnormal operator Y remain true for tuples; e.g. the existence and uniqueness of its minimal normal extension N, the spectral inclusion σ(N)⊂ σ(Y)-proved for n-tuples in [4] and generalized to infinite tuples in [5]. However, neither the invariant subspace theorem nor the spectral mapping theorem in the “strong form” as in [3] is known so far for subnormal tuples.

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 A characterization of subnormal operators

1984 ◽  
Vol 25 (1) ◽  
pp. 99-101 ◽  
Author(s):  
Alan Lambert
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Commutative Algebras ◽  
Binomial Expansion ◽  
Subnormal Operators

In this note a characterization of subnormality of operators on Hilbert space is given. The characterization is in terms of a sequence of polynomials in the operator and its adjoint reminiscent of the binomial expansion in commutative algebras. As such no external Hilbert spaces are needed, nor is it necessary to introduce forms dependent on arbitrary sequences of vectors from the Hilbert space.

On Quasisimilarity for Subnormal Operators, II

1982 ◽  
Vol 25 (1) ◽  
pp. 37-40 ◽  
Author(s):  
John B. Conway
Keyword(s):  
Subnormal Operator ◽  
Unilateral Shift ◽  
Closed Algebra ◽  
Subnormal Operators ◽  
Weak Star

AbstractLet S be a subnormal operator and let be the weak-star closed algebra generated by S and 1. An example of an irreducible cyclic subnormal operator S is found such that there is a T in with S and T quasisimilar but not unitarily equivalent. However, if S is the unilateral shift, T ∈ and S and T are quasisimilar, then S ≅ T.

scholarly journals Characterizations of Double Commutant Property on B H

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Chaoqun Chen ◽  
Fangyan Lu ◽  
Cuimei Cui ◽  
Ling Wang
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Necessary Condition ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Double Commutant ◽  
Sufficient And Necessary Condition

Let H be a complex Hilbert space. Denote by B H the algebra of all bounded linear operators on H . In this paper, we investigate the non-self-adjoint subalgebras of B H of the form T + B , where B is a block-closed bimodule over a masa and T is a subalgebra of the masa. We establish a sufficient and necessary condition such that the subalgebras of the form T + B has the double commutant property in some particular cases.

scholarly journals On the commutants modulo Cp of A2 and A3

1986 ◽  
Vol 41 (1) ◽  
pp. 47-50 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Fredholm Operator ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Subnormal Operators ◽  
Dimensional Hilbert Space

AbstractWe prove the following statements about bounded linear operators on a complex separable infinite dimensional Hilbert space. (1) Let A and B* be subnormal operators. If A2X = XB2 and A3X = XB3 for some operator X, then AX = XB. (2) Let A and B* be subnormal operators. If A2X – XB2 ∈ Cp and A3X – XB3 ∈ Cp for some operator X, then AX − XB ∈ C8p. (3) Let T be an operator such that 1 − T*T ∈ Cp for some p ≥1. If T2X − XT2 ∈ Cp and T3X – XT3 ∈ Cp for some operator X, then TX − XT ∈ Cp. (4) Let T be a semi-Fredholm operator with ind T < 0. If T2X − XT2 ∈ C2 and T3X − XT3 ∈ C2 for some operator X, then TX − XT ∈ C2.

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