Contractions Satisfying the Absolute Value Property

Let B(H) denote the algebra of operators on a complex Hilbert space H, and let U denote the class of operators which satisfy the absolute value condition . It is proved that if is a contraction, then either A has a nontrivial invariant subspace or A is a proper contraction and the nonnegative operator is strongly stable. A Putnam-Fuglede type commutativity theorem is proved for contractions A in , and it is shown that if normal subspaces of . It is proved that if are reducing, then every compact operator in the intersection of the weak closure of the range of the derivation with the commutant of A* is quasinilpotent.     

Inequalities of Putnam and Berger-Shaw forp-quasihyponormal operators

For an operatorT satisfying thatT ^*(T ^* T–TT ^*)T0, we shall show that and, moreover, tr itT isn-multicyclic.For an operatorT satisfying thatT ^* {(T ^* T) ^p –(TT ^*) ^p }T0 for somep (0, 1], we shall show that and, moreover, ifT isn-multicyclic.     

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

Aluthge Transforms of Complex Symmetric Operators

If denotes the polar decomposition of a bounded linear operator T, then the Aluthge transform of T is defined to be the operator . In this note we study the relationship between the Aluthge transform and the class of complex symmetric operators (T iscomplex symmetric if there exists a conjugate-linear, isometric involution so that T = CT*C). In this note we prove that: (1) the Aluthge transform of a complex symmetric operator is complex symmetric, (2) if T is complex symmetric, then and are unitarily equivalent, (3) if T is complex symmetric, then if and only if T is normal, (4) if and only if T ² = 0, and (5) every operator which satisfies T ² = 0 is necessarily complex symmetric. This work partially supported by National Science Foundation Grant DMS 0638789.     

Weyl’s Theorem for Algebraically Paranormal Operators

Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyls theorem holds for f(T) for every f $\in$ H((T)); (ii) a-Browders theorem holds for f(S) for every S $\prec$ T and f $\in$ H((S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Analysis of Non-normal Operators via Aluthge Transformation

Let T be a bounded linear operator on a complex Hilbert space . In this paper, we show that T has Bishops property () if and only if its Aluthge transformation has property (). As applications, we can obtain the following results. Every w-hyponormal operator has property (). Quasi-similar w-hyponormal operators have equal spectra and equal essential spectra. Moreover, in the last section, we consider Chs problem that whether the semi-hyponormality of T implies the spectral mapping theorem Re(T) = (Re T) or not.     

Orbits, weak orbits and local capacity of operators

LetT be an operator on a Banach spaceX. We give a survey of results concerning orbits {T ⁿ x:n=0,1,...} and weak orbits {T ⁿ x,x ^*:n=0,1,...} ofT wherexX andx ^*X ^*. Further we study the local capacity of operators and prove that there is a residual set of pointsxX with the property that the local capacity cap(T, x) is equal to the global capacity capT. This is an analogy to the corresponding result for the local spectral radius.The research was supported by the grant No. A1019801 of AV R.     

Riesz Idempotent and Algebraically M-hyponormal Operators

Let T be an M-hyponormal operator acting on infinite dimensional separable Hilbert space and let be the Riesz idempotent for λ₀, where D is a closed disk of center λ₀ which contains no other points of σ (T). In this note we show that E is self-adjoint and As an application, if T is an algebraically M-hyponormal operator then we prove : (i) Weyl’s theorem holds for f(T) for every (ii) a-Browder’s theorem holds for f(S) for every and f ∈ H(σ(S)); (iii) the the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Characterizations of logA≥logB and normaloid operators via Heinz inequality

In 1951, Heinz showed the following useful norm inequality:If A, B0and XB(H), then AXB ^r X^1–r A ^r XB ^r holds for r [0, 1]. In this paper, we shall show the following two applications of this inequality:Firstly, by using Furuta inequality, we shall show an extension of Cordes inequality. And we shall show a characterization of chaotic order (i.e., logAlogB) by a norm inequality.Secondly, we shall study the condition under which , where is Aluthge transformation ofT. Moreover we shall show a characterization of normaloid operators (i.e.,r(T)=T) via Aluthge transformation.     

On log-hyponormal operators

LetTB(H) be a bounded linear operator on a complex Hilbert spaceH.TB(H) is called a log-hyponormal operator itT is invertible and log (TT ^*)log (T ^* T). Since log: (0, )(–,) is operator monotone, for 0<p1, every invertiblep-hyponormal operatorT, i.e., (TT ^*) ^p (T ^* T) ^p , is log-hyponormal. LetT be a log-hyponormal operator with a polar decompositionT=U|T|. In this paper, we show that the Aluthge transform is . Moreover, ifmeas ((T))=0, thenT is normal. Also, we make a log-hyponormal operator which is notp-hyponormal for any 0<p.This research was supported by Grant-in-Aid Research No. 10640185     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

The C*-algebra generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C*-algebras associated with points and pairs We construct a symbol calculus for unital C*-algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈∂Π being the discontinuity points of coefficients. A symbol calculus for the C*- algebra and a Fredholm criterion for the operators are obtained. Finally, a C*-algebra isomorphism between the quotient algebra where is the ideal of compact operators, and its analogue for the unit disk is constructed.     

Putnam’s Inequality for log-Hyponormal Operators

Let T be a bounded linear operator on a complex Hilbert space H. T $/in$ B(H) is called a log-hyponormal operator if T is invertible and log (TT ^*) log (T ^* T). Since a function log : (0,) (-,) is operator monotone, every invertible p-hyponormal operator T, i.e., (TT ^*) ^p (T ^* T ^p is log-hyponormal for 0 < p 1. Putnams inequality for p-hyponormal operator T is the following:$ \| (T^*T)^p-(TT^*)^p \|\leq\frac{p}{\pi}\int\int_{\sigma(T)}r^{2p-1}drd\theta $.In this paper, we prove that if T is log-hyponormal, then$ \| log(T^*T)-log(TT^*) \|\leq\frac{1}{\pi}\int\int_{\sigma(T)}r^{-1}drd\theta $.     

Syndetically Hypercyclic Operators

Given a continuous linear operator T L(x) defined on a separable -space X, we will show that T satisfies the Hypercyclicity Criterion if and only if for any strictly increasing sequence of positive integers such that the sequence is hypercyclic. In contrast we will also prove that, for any hypercyclic vector x X of T, there exists a strictly increasing sequence such that and is somewhere dense, but not dense in X. That is, T and do not share the same hypercyclic vectors.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

Riesz Idempotent and Weyl’s Theorem for w-hyponormal Operators

Let T be a w-hyponormal operator on a Hilbert space H, its Aluthge transform, λ an isolated point of the spectrum of T, and E_λ and the Riesz idempotents, with respect to λ, of T and respectively. It is shown that Consequently, E_λ is self-adjoint, and if λ ≠ 0. Moreover, it is shown that Weyl’s theorem holds for f(T), where f ∈ H(σ (T)).     

The Aronszajn–Donoghue Theory for Rank One Perturbations of the $$\mathcal{H}_{-2} {\text{-Class}}$$

A singular rank one perturbation of a self-adjoint operator A in a Hilbert space is considered, where and but with the usual A–scale of Hilbert spaces. A modified version of the Aronszajn-Krein formula is given. It has the form where F denotes the regularized Borel transform of the scalar spectral measure of A associated with . Using this formula we develop a variant of the well known Aronszajn–Donoghue spectral theory for a general rank one perturbation of the class.Submitted: March 14, 2002 Revised: December 15, 2002     

Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

C*-Algebras of Integral Operators with Piecewise Slowly Oscillating Coefficients and Shifts Acting Freely

We establish a symbol calculus for the C*-subalgebra of generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators where is the Cauchy singular integral operator and The C*-algebra is invariant under the transformations