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In the present paper, we characterize the Fredholmness of Toeplitz pairs on Hardy space over the bidisk with the bounded holomorphic symbols, and hence, we obtain the index formula for such Toeplitz pairs. The key to obtain the Fredholmness of such Toeplitz pairs is the $L^p$ solution of Corona Problem over $\mathbb {D}^2$.

Summary

This chapter is an introduction to the geometric perspective of tempered representations of connected reductive Lie groups via Dirac induction and the Connes–Kasparov isomorphism. We start with discussing Dirac operators in spin geometry. We then recall some classical Dirac operators on homogeneous spaces arising in representation theory. Next, we introduce the theory of higher index to be used as a language to formulate the Connes–Kasparov conjecture. In Section 3.4, we introduce Dirac induction, the Connes–Kasparov conjecture, and survey a couple of ways for proving it. The final section is devoted to the harmonic-analytic approach to the Connes–Kasparov isomorphism illustrated by the example of connected semisimple groups.

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Fredholm index of Toeplitz pairs with $H^{\infty }$ symbols

3 - Dirac Operators and Representation Theory

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2 results

Fredholm index of Toeplitz pairs with $H^{\infty }$ symbols

3 - Dirac Operators and Representation Theory

Summary