THE CLOSED RANGE POINTS AND QUASISIMILARITY OF WEIGHTED SHIFT OPERATORS OF DEGREE K

Lambert showed in 1970 that two quasisimilar injective unilateral weighted shifts must be similar and hence have the same dosed range points. But whether the conclusion is true for injective bilateral weighted shift operators has not been proved yet. In this note we answer the question affirmatively with a stronger result. We prove that two quasisimilar injective bilat-

Quasisimilarity of Cowen-Douglas operators

This paper shows that every operator which is quasisimilar to strongly irreducible Cowen-Douglas operators is still strongly irreducible. This result answers a question posted by Davidson and Herrero (ref. [1]).

The Properties of k-quasi-＊-A（n） Operator

An operator T is called k-quasi-*-A(n) operator, if T^(*k)|T^(1+n)|^(2/(1+n))T^k ≥T^(*k)|T~* |~2T^k , k ∈ Z, which is a generalization of quasi-*-A(n) operator. In this paper we prove some properties of k-quasi-*-A(n) operator, such as, if T is a k-quasi-*-A(n) operator and N(T )■N(T~* ), then its point spectrum and joint point spectrum are identical. Using these results, we also prove that if T is a k-quasi-*-A(n) operator and N(T )■N(T ), then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.

SOME RESULTS ON THE QU OF OPERATORS

B. Sz-Nagy and C. Foias introduced the concept of quasisimilarity in 1967. Since then, the quasisimilarity of operators has become an important research problem in the theory of operators and a lot of attractive results have been obtained. But, as regards intersection relations between some subsets of the spectrum σ(A)and those of the spectrum σ(B), where A and B are quasisimilar operators, comparatively few results have been delivered.

COMPONENTS OF THE ESSENTIAL SPECTRA OF QUASISIMILAR OPERATORS

Let A and B be quasisimilar operators. We describe refinedly the intersection relationsof the components of various essential spectra of A with various subsets of the essentialspectrum of B, and give an affirmative answer to a question posed by L. A. Fialkow.

Intersection Relations Between the Spectra of Quasisimilar Operators

Let X denote an infinite dimensional complex Banach space and L(X) denote the set of all the bounded linear operators on X. For A∈L(X), B∈L (Y), A and B are quasisimilar (written as A (?) B) if there exist P:X→Y, Q: Y→X, where P and Q are bounded linear, injective and dense-ranged such that PA=BP, QB=AQ. T. B.

THE ESSENTIAL SPECTRA OF OPERATORS QUASISIMILAR TO SUBNORMAL OPERATORS

Yang Liming showed in 1988 that if S is a subnormal operator, T is a hyponormal operator and T and S are quasisimilar, then σ_e(S)(?) σ_e(T), and hence he deduced the conclusion that two quasisimilar subnormal operators have equal essential spectra. This is an important result in the theory of quasisimilarity. In this note we improve Yang’s method to show that if S or S is subnormal, T is a bounded linear operator and T and S are quasisimilar, then σ_e(S) (?) σ_e(T).

Strongly Irreducible Operators on Banach Spaces

This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w＊-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.