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Abstract.A truncated Toeplitz operator Aϕ : K Θ → K Θ is the compression of a Toeplitz operator Tϕ :For Θ inner, let T Θ denote the set of all bounded truncated Toeplitz operators on K Θ . Our main result is a necessary and sufficient condition on inner functions Θ 1 and Θ 2 which guarantees that T Θ 1 and T Θ 2 are spatially isomorphic. (i.e., U T Θ 1 = T Θ 2 U for some unitary U : K Θ 1 → K Θ 2 ). We also study operators which are unitarily equivalent to truncated Toeplitz operators and we prove that every operator on a finite dimensional Hilbert space is similar to a truncated Toeplitz operator.

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Abstract. In this paper, we study the matrix representations of compressions of Toeplitz operators to the finite dimensional model spaces H 2 BH 2 , 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 H 2 BH 2 is the compression of a Toeplitz operator. This result complements a related result of Sarason [6]. (2000): Toeplitz operators, model spaces, Clark operators, matrix representations. Mathematics subject classification

Abstract. We characterize the compact composition operators on BMOA, the space consisting of those holomorphic functions on the open unit disk U that are Poisson integrals of functions on ∂U , that have bounded mean oscillation. We then use our characterization to show that compactness of a composition operator on BMOA implies its compactness on the Hardy spaces (a simple example shows the converse does not hold). We also explore how compactness of the composition operator C φ : BMOA → BMOA relates to the shape of φ(U ) near ∂U , introducing the notion of mean order of contact. Finally, we discuss the relationships among compactness conditions for composition operators on BMOA, VMOA, and the big and little Bloch spaces.

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