# Journal of Operator Theory

Volume 49, Issue 1, Winter 2003 pp. 115-141.

Analogues of composition operators on non-commutative $H^p$-spaces**Authors**: L.E. Labuschagne

**Author institution:**Department of Mathematics and Applied Mathematics, University of Pretoria, 0002 Pretoria, South Africa

**Summary:**We briefly review the theory of non-commutative $H^p$-spaces and suggest a possible non-commutative analogue of the disc algebra. We then pass to the theory of composition operators and proceed to identify some of the basic algebraic principles underlying the theory of these operators on classical Hardy spaces. This framework is then generalised to the non-commutative setting. Here we succeed in describing what may be regarded as non-commutative analogues of composition operators. Building on these ideas it is shown that even in this very general context one yet finds what effectively constitutes operator theoretic remnants of the Littlewood Subordination Principle (see Proposition 2.4 and Theorem 4.12). In conclusion we investigate the connection between linear isometries on non-commutative $H^p$ spaces and analogues of composition operators.

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