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General Toeplitz kernels and
$(X,Y)$-invariance
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 27 March 2023, pp. 680-706
- Print publication:
- April 2024
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- Article
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Motivated by the near invariance of model spaces for the backward shift, we introduce a general notion of
$(X,Y)$-invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).
