Analytic theory of subnormal operators

LetS be a pure subnormal operator and (·) be its mosaic introduced in [13]. In this paper, the author proves that rank (z)<+ if [S ^*,S] is compact and rank (z)=–g(z) whereg(·) is the Pincus principal function if [S ^*,S] is in the trace class. Also a result is given about the form of pure and irreducible subnormal operators with finite rank self-commutators.This work is supported in part by a NSF grant.     

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.     

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.

     

Cellular-indecomposable subnormal operators

A bounded operator T is cellular-indecomposable if LnM{0} whenever L and M are any two nonzero invariant subspaces for T. We show that any such subnormal operator has a cyclic normal extension and is unitarily equivalent modulo the compact operators to an analytic Toeplitz operator whose symbol is a weak-star generator of H.Dedicated to the memory of James P. WilliamsThis work was supported in part by a grant from the National Science Foundation.     

Some invariant subspaces for subnormal operators

A theorem of D.E. Sarason is used to show that all subnormal operators have nontrivial invariant subspaces if some very special subnormal operators have them. It is then shown that these special subnormal operators as well as certain other operators do in fact have nontrivial invariant subspaces.     

An integrality theorem for subnormal operators

In a recent paper we conjectured that the principal function of a cyclic subnormal operator T is a.e. equal to the negative of a characteristic function. We showed that this was true in a variety of cases - including the general arc length Swiss Cheese.Now we prove stronger results. The conjecture is a consequence of:The principal function of a subnormal operator with trace class self-commutator assumes a.e. nonpositive integer values.It is an interesting fact that this integrality is a basic geometric property of subnormal operators and is not associated with any smoothness or "thinness" of the essential spectrum of T.This result is actually a simple corollary of a much more basic fact:The mosaic of a subnormal operator with trace class self-commutator is projection valued a.e.We have long known that the mosaic is a complete unitary invariant for T. Thus, this theorem establishes a map z Range B(z) which associates a subspace of Hilbert space with almost every point of the plane; and this generalized bundle completely characterizes the subnormal operator T. If T is cyclic then its mosaic B(·) is a.e. either the zero operator or a rank one projection.     

Hyperinvariant subspaces for some subnormal operators

In this article we employ a technique originated by Enflo in 1998 and later modified by the authors to study the hyperinvariant subspace problem for subnormal operators. We show that every ``normalized'subnormal operator such that either does not converge in the SOT to the identity operator or does not converge in the SOT to zero has a nontrivial hyperinvariant subspace.

     

Cellular-indecomposable subnormal operators II

This work continues that begun in [9]. Our investigation has led us to the following conjecture: a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator of H. In this paper some particular cases of the conjecture are verified.This work was supported in part by a grant from the National Science Foundation.     

On self-dual subnormal operators

We improve the result of C. C. Huang about self-dual subnormal operators, and consider the converse of this result.     

Algebras of subnormal operators

The paper deals with the following: (I) If S is a subnormal operator on $\text{H}$, then $\text{Ol}$(S) = $\text{W}$(S) = Alg Lat S. (II) If L ∈ ($\text{Ol}$(S), σ-wot)¹, then there exist vectors a and b in $\text{H}$ such that L(T) = 〈Ta, b〉 for every T in $\text{Ol}$. (III) In addition to I the map i(T) = T is a homeomorphism from ($\text{Ol}$, σ-wot) onto ($\text{W}$(S), wot). (IV) If S is not a reductive normal operator, then there exists a cyclic invariant subspace for S that has an open set of bounded point evaluations. (This open set can be constructed to be as large as possible.)     

Cellular-indecomposable subnormal operators III

A bounded operatorT is called cellular-indecomposable ifL M {0} wheneverL andM are nonzero invariant subspaces forT. We prove that a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator ofH . This completes our previous work [5], [6].     

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.     

The generalised wold decomposition for subnormal operators

Subnormal operatorsS with the spectrum of the minimal normal extension contained in the boundary of (S) are studied. Under certain geometric assumptions it is shown that (up to unitary equivalence)S is the orthogonal sum of a normal operator and of the multiplication by the independent variablez on the Hardy spaceH ² [E] of a certain flat unitary bundleE over the interior of (S). This extends the results of Abrahamse and Douglas [1], [2].     

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.

     

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.     

Subnormal roots of subnormal operators

Suppose that S is a subnormal operator and that S has a square root. Must S have a subnormal square root? We give two examples which answer this question in the negative.     

Tensor products of subnormal operators

We shall use a -algebra approach to study operators of the form where is subnormal and is normal. We shall determine the spectral properties for these operators, and find the minimal normal extension and the dual operator. We also give a necessary condition for to contain a compact operator and a sufficient condition for the algebraic equivalence of and .

We also consider the existence of a homomorphism satisfying . We shall characterize the operators such that exists for every operator .

The problem of when is unitarily equivalent to is considered. Complete results are given when and are positive operators with finite multiplicity functions and has compact self-commutator. Some examples are also given.

     

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.