Toeplitz operators and pseudo-extensions

Abstract

There are three main results in this paper. First, we find an easily computable and simple condition which is necessary and sufficient for a commuting tuple of contractions to possess a non-zero Toeplitz operator. This condition is just that the adjoint of the product of the contractions is not pure. On one hand this brings out the importance of the product of the contractions and on the other hand, the non-pureness turns out to be equivalent to the existence of a pseudo-extension to a tuple of commuting unitaries. The second main result is a commutant pseudo-extension theorem obtained by studying the unique canonical unitary pseudo-extension of a tuple of commuting contractions. The third one is about the C * -algebra generated by the Toeplitz operators determined by a commuting tuple of contractions. With the help of a special completely positive map, a different proof of the existence of the unique canonical unitary pseudo-extension is given.

Factorization of Toeplitz operators

“…We begin with the equivalent criteria for the space of all T -Toeplitz operators T (T ) to be non-trivial. This result is proved in Theorem 1.3 of [6] and can also be proved using results in [20]. A similar kind of result is also proved in an another context in [7].…”

“…, M z d ), the d-tuple of multi-shifts on the Hardy space H 2 (D d ) over the polydisc D d , then T -Toeplitz operators are precisely the Toeplitz operators on H 2 (D d ) [17]. It has been shown in [6] that the existence of non-zero T -Toeplitz operators is related to isometric pseudo-extensions of T . Definition 1.1.…”

“…, V d ) is an d-tuple of commuting isometries (unitaries). An isometric (unitary) pseudo-extension (J , K, V ) of T is said to be minimal if K is the smallest joint invariant (reducing) space for V containing J H. A minimal isometric or unitary pseudoextension [6], it is proved that a non-zero T -Toeplitz operator exists if and only if T has a canonical isometric pseudo-extension (J , K, V ). In fact, if (J , K, V ) is any isometric pseudo-extension of T , then it is easy to see that J * J is a non-zero contractive positive T -Toeplitz operator.…”