ACTA issues

## Remarks, examples and spectral properties of generalized Toeplitz operators

Carmen H. Mancera, Pedro J. Paúl

Acta Sci. Math. (Szeged) 66:3-4(2000), 737-753
2765/2009

 Abstract. An operator $X\colon{\cal H}_1 \to{\cal H}_2$ is said to be a generalized Toeplitz operator with respect to given contractions $T_1$ and $T_2$ if $X=T_2XT_1^*$. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foiaş, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give some clarifying examples and a new characterization of analytic Toeplitz operators that complement the work done by Pták and Vrbová, as well as some spectral properties of generalized Toeplitz operators that complement the work done by Sz.-Nagy and Foiaş. As a by-product we prove that the spectrum of a function $\phi\in H^\infty$ equals the approximate point spectrum of its Toeplitz operator. AMS Subject Classification (1991): 47B35 Keyword(s): Toeplitz operators, spectral properties, minimal isometric dilation Received December 21, 1998, and in revised form November 18, 1999. (Registered under 2765/2009.)