Toeplitz Operators

If $\varphi$ is a bounded measurable function on the unit circle, then the multiplication induced by $\varphi$ on $L^2$ (with respect to normalized Lebesgue measure $\mu$) is sometimes called the Laurent operator induced by $\varphi$, in symbols $L_\varphi$. A Laurent matrix is a (bilaterally) infinite matrix $\langle \lambda_{ij}\rangle$ such that $\lambda_{i+1,j+1} = \lambda_{ij}$ for all $i$ and $j$ $(=0, \pm 1, \pm 2, \ldots )$ — in words, a Laurent matrix is one all of whose diagonals (parallel to the main diagonal) are constants. The basis in question is $e_n(z) = z^n$, $n = 0, \pm 1, \pm 2, \ldots$.

Laurent operators (multiplications) are distinguished operators on $L^2$ (of the unit circle), and $H^2$ is a distinguished subspace of $L^2$; something interesting is bound to happen if Laurent operators are compressed to $H^2$. If $P$ is the projection from $L^2$ onto $H^2$, and if $\varphi$ is a bounded measurable function, then the Toeplitz operator $T_\varphi$ induced by $\varphi$ is defined by $T_\varphi f = P(\varphi \cdot f)$ for all $f$ in $H^2$. The simplest non-trivial example of a Laurent operator is the bilateral shift $W \, (= L_{e_1})$; correspondingly, the simplest non-trivial example of a Toeplitz operator is the unilateral shift $U \, (= T_{e_1})$.

A Toeplitz matrix is a (unilaterally) infinite matrix $\langle \lambda_{ij}\rangle$ such that $\lambda_{i+1,j+1} = \lambda_{ij}$ for all $i$ and $j$ $(=0, 1, 2, \ldots )$ — again, one all of whose diagonals (parallel to the main diagonal) are constants; the difference from a Laurent matrix is that for Toeplitz matrices the indices go forward only, not both ways.

The Toeplitz mapping $\varphi \mapsto T_\varphi$ from functions to operators is not multiplicative; the question is whether it is at least multiplicative modulo compact operators.

Recall that associated with each $\varphi$ in $H^\infty$ there is a function $\tilde\varphi$ analytic in the open unit disc $D$ (Problem 35). The spectral behavior of $T_\varphi$ is influenced by the complex analytic behavior of $\tilde\varphi$ rather than by the merely set-theoretic behavior of $\varphi$. Here is still another way to express the result of this problem: if $\varphi \in L^\infty$, then the spectrum of $L_\varphi$ is the essential range of $\varphi$; if $\varphi \in H^\infty$, then the spectrum of $T_\varphi$ is what may be called the essential range of $\tilde\varphi$.