Abstract. The Hilbert space operator $C$ is called $T$-Toeplitz if the equation $T^*CT=r(T)^2C$ holds, where $r(T)$ denotes the spectral radius of $T$. The set of all $T$-Toeplitz operators is studied, for an arbitrary bounded, linear operator $T$. It turns out that a satisfactory symbolic calculus can be given, if $T$ has a regular norm-sequence $\{\|T^n\|\} _{n=1}^\infty $. A projection mapping onto the set of $T$-Toeplitz operators is constructed, the spectral properties of the symbolic calculus are examined, and invariant subspace theorems for $T$ are derived from the study of $T$-Toeplitz operators. These investigations are also extended to the case when $T$ is replaced by a representation $\rho $ of an abelian semigroup.

AMS Subject Classification (1991): 47A62, 47A10, 47A15, 47B35

Received September 6, 2001. (Registered under 2847/2009.)