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Journal of Operator Theory

Volume 40, Issue 1, Summer 1998  pp. 185-207.

Norms of some singular integral operators and their inverse operators

Authors Takahiko Nakazi (1), and Takanori Yamamoto (2)
Author institution: (1) Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060, Japan
(2) Department of Mathematics, Hokkai-Gakuen University, Sapporo 062, Japan

Summary:  Let $\alpha$ and $\beta$ be bounded measurable functions on the unit circle $\bbb T$. Then the singular integral operator $S_{\alpha,\beta}$ is defined by $S_{\alpha,\beta}f = \alpha P_+ f + \beta P_- f$, $(f \in L^2({\bbb T}))$ where $P_+$ is an analytic projection and $P_-$ is a co-analytic projection. In this paper, the norms of $S_{\alpha,\beta}$ and its inverse operator on the Hilbert space $L^2({\bbb T})$ are calculated in general, using $\alpha, \beta$ and $\alpha\bar{\beta} + H^\infty$. Moreover, the relations between these and the norms of Hankel operators are established. As an application, in some special case in which $\alpha$ and $\beta$ are nonconstant functions, the norm of $S_{\alpha,\beta}$ is calculated in a completely explicit form. If $\alpha$ and $\beta$ are constant functions, then it is well known that the norm of $S_{\alpha,\beta}$ on $L^2({\bbb T})$ is equal to $\max \left\{ |\alpha|, |\beta| \right\}$. If $\alpha$ and $\beta$ are nonzero constant functions, then it is also known that $S_{\alpha,\beta}$ on $L^2({\bbb T})$ has an inverse operator $S_{{\alpha}^{-1},{\beta}^{-1}}$ whose norm is equal to $\max \left\{ {|\alpha|}^{-1}, {|\beta|}^{-1} \right\}$.

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