Warren R. Wogen

Some New Classes of Complex Symmetric Operators

We say that an operator T E B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C: ℌ→ℌ so that T = CT * C. We prove that binormal operators, operators that are algebraic of degree two (including all idempotents), and large classes of rank-one perturbations of normal operators are complex symmetric. From an abstract viewpoint, these results explain why the compressed shift and Volterra integration operator are complex symmetric. Finally, we attempt to describe all complex symmetric partial isometries, obtaining the sharpest possible statement given only the data (dim ker T, dim ker T*).

Complex symmetric partial isometries

An operator T∈B(H) is complex symmetric if there exists a conjugate-linear, isometric involution C:H→H so that T=CT∗C. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension ⩽4 is complex symmetric.

Reflexivity properties of T ⊕ 0

A construction is given of a reflexive operator T acting on a separable Hilbert space H with the property that the direct sum T ⊕ 0 fails to be reflexive. This construction is then used to provide solutions to several other problems which have been studied concerning the direct-sum splitting of operator algebras, Scott Browns technique, the theory of bitriangular operators, and parareflexivity.

On Boundedness of Composition Operators on H 2 (B 2 )

Composition operators on the Hardy space H2 of the ball in C2 are studied. Some sufficient conditions are given for a composition operator to be bounded. A class of inner mappings is given which induces isometric composition operators. Another class of inner mappings is shown to induce unbounded composition operators.

C* -algebras Generated by Truncated Toeplitz Operators

We obtain an analogue of Coburn’s description of the Toeplitz algebra in the setting of truncated Toeplitz operators. As a byproduct, we provide several examples of complex symmetric operators which are not unitarily equivalent to truncated Toeplitz operators having continuous symbols.

Extensions of normal operators

We prove that every one dimensional extension of a separably acting normal operator has a cyclic commutant, and that every non-algebraic normal operator has a two-dimensional extension which fails to have a cyclic commutant. Contrasting this, we prove that ifT is an extension of a normal operator by an algebraic operator then the weakly closed algebraW(T) has a separating vector.

On cyclicity of commutants

In this note an example is given of a Hilbert space operator whose commutant is not finitely cyclic. The example settles in the negative a conjecture of D. A. Herrero.

Separating versus strictly separating vectors

Let S be a linear manifold of Banach space operators which is closed in the strong operator topology. Existence of a disjoint pair of separating vectors does not guarantee reflexivity of S, but S must be reflexive if one of these vectors is strictly separating. S must also be reflexive if all non–zero linear combinations of some independent pair of vectors strictly separate S.

Extensions of bitriangular operators

We prove that every one-dimensional extension of a bitriangular operator has a cyclic commutant. We also prove that ifT is an extension of a bitriangular operator by an algebraic operator, then the weakly closed algebraW(T) generated byT has a separating vector.

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.