Essentially subnormal operators

An operator is essentially subnormal if its image in the Calkin algebra is subnormal. We shall characterize the essentially subnormal operators as those operators with an essentially normal extension. In fact, it is shown that an essentially subnormal operator has an extension of the form ``normal plus compact'.

The essential normal spectrum is defined and is used to characterize the essential isometries. It is shown that every essentially subnormal operator may be decomposed as the direct sum of a subnormal operator and some irreducible essentially subnormal operators. An essential version of Putnam's Inequality is proven for these operators. Also, it is shown that essential normality is a similarity invariant within the class of essentially subnormal operators. The class of essentially hyponormal operators is also briefly discussed and several examples of essentially subnormal operators are given.

     

Separating classes of composition operators via subnormal condition

Several classes have been considered to study the weak subnormalities of Hilbert space operators. One of them is -hypnormality, which comes from the Bram-Halmos criterion for subnormal operators. In this note we consider -hyponormality, which is the parallel version corresponding to the Embry characterization for subnormal operators. We characterize -hyponormality of composition operators via -th Radon-Nikodym derivatives and present some examples to distinguish the classes.

     

Seminormality of operators from their tensor product

The question of seminormality of tensor products of nonzero
bounded linear operators on Hilbert spaces is investigated. It is shown that is subnormal if and only if so are and .

     

On quasi-similarity of subnormal operators

For a compact subset K in the complex plane, let Rat(K) denote the set of the rational functions with poles off K. Given a finite positive measure with support contained in K, let R2(K,v) denote the closure of Rat(K) in L2(v) and let Sv denote the operator of multiplication by the independent variable z on R2(K, v), that is, Svf = zf for every f∈R2(K, v). SupposeΩis a bounded open subset in the complex plane whose complement has finitely many components and suppose Rat(Ω) is dense in the Hardy space H2(Ω). Letσdenote a harmonic measure forΩ. In this work, we characterize all subnormal operators quasi-similar to Sσ, the operators of the multiplication by z on R2(Ω,σ). We show that for a given v supported onΩ, Sv is quasi-similar to Sσif and only if v/■Ω■σ and log(dv/dσ)∈L1(σ). Our result extends a well-known result of Clary on the unit disk.     

On the spectral picture of an irreducible subnormal operator II

In this paper we show that the spectral picture of an irreducible subnormal operator may be arbitrarily prescribed subject only to certain natural necessary conditions. This completes work begun by the second author.

     

On unbounded subnormal operators

A minimal normal extension of unbounded subnormal operators is established and characterized and spectral inclusion theorem is proved. An inverse Cayley transform is constructed to obtain a closed unbounded subnormal operator from a bounded one. Two classes of unbounded subnormals viz analytic Toeplitz operators and Bergman operators are exhibited.     

Backward extensions of subnormal operators

The concept of backward extension for subnormal weighted shifts is generalized to arbitrary subnormal operators. Several differences and similarities in these contexts are explored, with emphasis on the structure of the underlying measures.

     

Jointly hyponormal pairs of commuting subnormal operators need not be jointly subnormal

We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.

     

Tensor products over abelian Walgebras

Tensor products of Calgebras over an abelian Walgebra are studied. The minimal Cnorm on is shown to be just the quotient of the minimal Cnorm on if or is exact.

     

On subnormal operators

LetS be a pure subnormal operator such thatC*(S), theC*-algebra generated byS, is generated by a unilateral shiftU of multiplicity 1. We obtain conditions under which 5 is unitarily equivalent toα + βU, α andβ being scalars orS hasC*-spectral inclusion property. It is also proved that if in addition,S hasC*-spectral inclusion property, then so does its dualT andC*(T) is generated by a unilateral shift of multiplicity 1. Finally, a characterization of quasinormal operators among pure subnormal operators is obtained.     

L—fuzzy模范畴的张量积与张量函子

本文在Ｌ—ｆｕｚｚｙ模范畴中，建立了相应的张量积，给出了它的结构性、存在性与唯一性定理，并讨论了张量函子与Ｈｏｍ函子的伴随性。所得结果为通常张量积的“良好推广”（ｇｏｏｄｅｘｔｅｎｓｉｏｎ）。     

Weyl's theorem, tensor products and multiplication operators

The transmission of “Weyl's theorem” from operators on Banach spaces to their tensor products, and also to their associated multiplication operators, is deconstructed.     

On the injective norm and characterization of some subclasses of normal operators by inequalities or equalities

Let B(H) be the C^*-algebra of all bounded linear operators acting on a complex Hilbert space H. In this note, we shall show that if S is an invertible normal operator in B(H) the following estimation holds

     

On products of unbounded operators

A symmetric operator X^ is attached to each operator X that leaves the domain of a given positive operator A invariant and makes the product AX symmetric. Some spectral properties of X^ are derived from those of X and, as a consequence, various conditions ensuring positivity of products of the form AX ₁ ... X _n are proved. The question of ^-complete positivity of the mapping p → Ap(X ₁,...,X _n) defined on complex polynomials in n variables is investigated. It is shown that the set ω is related to the McIntosh-Pryde joint spectrum of (X ₁,...,X _n) in case all the operators A, X ₁,...,X _n are bounded. Examples illustrating the theme of the paper are included. This revised version was published online in June 2006 with corrections to the Cover Date.     

An asymmetric Putnam-Fuglede theorem for unbounded operators

The intertwining relations between cosubnormal and closed hyponormal (resp. cohyponormal and closed subnormal) operators are studied. In particular, an asymmetric Putnam-Fuglede theorem for unbounded operators is proved.

     

良序套张量积的代数中的强不可约算子

设Ｈ为复的可分无限维Ｈｉｌｂｅｒｔ空间，称有界线性算子Ｔ为强不可约的，如果与Ｔ可交换的幂等算子只有０和Ｉ．王宗尧、蒋春澜、纪有清等人证明了在任何一个套的套代数中都存在大量的强不可约算子，并且找到了它们的酉轨道闭包．本文考虑有限个套的张量积的代数中强不可约算子的存在性问题。证明了：对复平面上任何一个连通完备集σ、总存在一个对角算子Ｎ和它的一个范数可以任意小的紧摄动Ｔ＝Ｘ＋Ｋ，使得Ｔ是一个强不可约算子、Ｔ在有限个良序套的张量积的代数中，并且σ（Ｔ）＝σｌｒｅ（Ｔ）＝σ（Ｎ）＝σｌｒｅ（Ｎ）＝σ进一步，文章还对具有单点谱的算子和良序套与正交补为良序套的张量积的代数进行了讨论，得到了一些结果．     

Positive definiteness and commutativity of operators

It is shown that an -tuple of bounded linear operators on a complex Hilbert space, which is positive definite in the sense of Halmos, must be commutative. Some generalizations of this result to the case of pairs of unbounded operators are obtained.

     

A local lifting theorem for subnormal operators

Criteria for the existence of lifts of operators intertwining subnormal operators are established. The main result of the paper reduces lifting questions for general subnormal operators to questions about lifts of cyclic subnormal operators. It is shown that in general the existence of local lifts (i.e. those coming from cyclic parts) for a pair of subnormal operators does not imply the existence of a global lift. However this is the case when minimal normal extensions of subnormal operators in question are star-cyclic.