Similarities involving bounded operators

Similarities involving bounded operators

Quasi-similarity of unbounded operators

In this section, we generalized some notions of bounded operators to unbounded operators on Hilbert space, and we extend some familiar results on quasi-similar bounded operators to unbounded operators. Defnition 4.2.1 (Quasi-similarity) Let T and S be densely defned linear operators in Hilbert spaces H1 and H2, respectively. If there exist quasi-invertible operators XT S from H1 into H2 and XST from H2 into H1 such that XT ST SXT S and XST S T XST then we say that T is quasi-similar to S and is denoted by T S: emark 4.2.2 if T S; then S

Defnition 4.2.3 A densely defned operator T is said to be hyponormal if:

a densely defned linear operator T in a Hilbert space H is said to be subnormal if there exist a Hilbert space K containing H as a closed subspace and a normal operator N in K such that D (T) D (N) and T x = Nx for all x 2 D (T):

Proposition 4.2.4 A subnormal operator is hyponormal

Proof. Suppose T is subnormal operator in a Hilbert space H with normal extension N in K; and let us take x 2 D (T); for y 2 D (T); one has hT x; yiH = hNx; yiK = hx; N yiK 86hence y 2 D (T):Moreover if P is the orthogonal projection of K onto H; then the above equalities implies that for all x 2 D (T); thus T is hyponormal.
The following Theorem is an unbounded version of Claryís result [Cla75] on unbounded operators. Theorem 4.2.5 ([OS89]) Let T be a closed hyponormal operator in a Hilbert space H and let S be a closed densely defned operator in Hilbert space K: If there exists a linear bounded transformation X with dense range from K to H such that XS T X; then the spectrum of T is contained in that of S: Corollary 4.2.6 Quasi-similar closed hyponormal operators have equal spectra.
The next Theorem is a generalization of Douglasí result in [Dou69] to unbounded operators. Theorem 4.2.7 ([OS89]) Let T and S be normal operators. If T and S are quasisimilar, then T and S are unitarily equivalent. Proof. Let T1(resp. S1) be the closure of T +T.

Unbounded operators similar to their adjoints

Theorem 4.4.1 ([DM17]) Let S be a bounded operator on a C-Hilbert space H such that 0 2= W (S). Let T be an unbounded and closed hyponormal operator with domain D(T) H. If ST T S, then T is self-adjoint. To prove the theorem stated above we need the following results. Lemma 4.4.2 Let T be a densely defned and closed operator in a Hilbert space H, with domain D(T) H. Assume that for some k > 0, Then ran(T) is closed. Proposition 4.4.3 Let T be an unbounded, closed and hyponormal operator in some Hilbert space H. Then W(T) co (T), where co (T) denotes the the convex hull of Proof. The proof is divided into three claims.

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Table des matières

Introduction vi
1 Essential background
1.1 Banach algebra
1.1.1 Introduction
1.1.2 Basic properties of spectra-algebra
1.3 Bounded operators
1.3.1 Defnitions and properties
1.3.2 Approximate point spectrum
1.3.3 Resolutions of the identity
1.4 Polar decomposition of an operator
1.4.1 Isometry and partial isometry
1.4.2 Polar decomposition of an operator
1.5 Positive operators
1.6 Numerical range
2 Non-normal operator classes
2.1 Compact operators
2.2 Hyponormal operators
2.2.1 Defnitions and properties
i2.2.2 Some conditions implying normality or self-adjointness
2.2.3 p-Hyponormal operators
2.3 Normaloid operators
2.4 Paranormal operators
2.4.1 Defnitions and properties
2.5 Convexoid operators
2.5.1 Examples
2.6 Class A operators
2.6.1 Quasi-class A operators
3 Similarities involving bounded operators
3.1 Introduction
3.2 Operators similar to their adjoints
3.2.1 Conditions implying self-adjointness of operators
3.2.2 Operators similar to self-adjoint ones
3.3 Operators with inverses similar to their adjoints
3.3.1 Operators similar to unitary ones
3.3.2 Operators with left inverses similar to their adjoints
3.4 Similarities involving normal operators
3.5 Quasi-similarity of operators
4 Similarities involving unbounded operators
4.1 Preliminaries
4.1.1 Adjoint Operators
4.1.2 Self-adjoint Operators
4.2 Quasi-similarity of unbounded operators
4.3 Similarities involving unbounded normal operators
ii4.4 Unbounded operators similar to their adjoints
Bibliography