Complex symmetric partial isometries

An operator T \in B(\h) is complex symmetric if there exists a conjugate-linear, isometric involution C:\h\to\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 $\leq 4$ is complex symmetric.Comment: 9 page

Reflexivity properties of T ⊕ 0

AbstractA 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 Brown's technique, the theory of bitriangular operators, and parareflexivity

C∗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.Comment: 12 page

Common Cyclic Vectors for Normal Operators

If μis a finite compactly supported measure on C, then the set Sμ of multiplication operators Mᵩ : L2 (μ) --\u3e L2 (μ), Mᵩ f = ᵩ f, where ᵩ ϵ L ∞ (μ) is injective on a set of full μ measure, is the complete set of cyclic multiplication operators on L2 (μ) In this paper, we explore the question as to whether or not Sμ has a common cyclic vecto

Spatial isomorphisms of algebras of truncated Toeplitz operators

We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.Comment: 24 page

Common Cyclic Vectors for Unitary Operators

In this paper, we determine whether or not certain natural classes of unitary multiplication operators on L2(dƟ) have common cyclic vectors. For some classes which have common cyclic vectors, we obtain a classification of these vectors

Truncated Toeplitz Operators on Finite Dimensional Spaces

In this paper, we study the matrix representations of compressions of Toeplitz operators to the finite dimensional model spaces H2ƟBH2, where B is a finite Blaschke product. In particular, we determine necessary and sufficient conditions - in terms of the matrix representation - of when a linear transformation on H2ƟBH2 is the compression of a Toeplitz operator. This result complements a related result of Sarason [6]

SOME NEW CLASSES OF COMPLEX SYMMETRIC OPERATORS

Abstract. We say that 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 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 ∗). 1