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• W4220900999 abstract "Abstract This paper studies matrix-valued truncated Toeplitz operators, which are a vectorial generalisation of truncated Toeplitz operators. It is demonstrated that, although there exist matrix-valued truncated Toeplitz operators without a matrix symbol in $$L^p$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> for any $$p in (2, infty ]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , there is a wide class of matrix-valued truncated Toeplitz operators which possess a matrix symbol in $$L^p$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> for some $$p in (2, infty ]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . In the case when the matrix-valued truncated Toeplitz operator has a symbol in $$L^p$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> for some $$p in (2, infty ]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , an approach is developed which bypasses some of the technical difficulties which arise when dealing with problems concerning matrix-valued truncated Toeplitz operators with unbounded symbols. Using this new approach, two new notable results are obtained. The kernel of the matrix-valued truncated Toeplitz operator is expressed as an isometric image of an $$S^*$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>S</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> -invariant subspace. Also, a Toeplitz operator is constructed which is equivalent after extension to the matrix-valued truncated Toeplitz operator. In a different yet overlapping vein, it is also shown that multidimensional analogues of the truncated Wiener–Hopf operators are unitarily equivalent to certain matrix-valued truncated Toeplitz operators." @default.
• W4220900999 created "2022-04-03" @default.
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• W4220900999 date "2022-01-29" @default.
• W4220900999 modified "2023-10-13" @default.
• W4220900999 title "Matrix-Valued Truncated Toeplitz Operators: Unbounded Symbols, Kernels and Equivalence After Extension" @default.
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• W4220900999 doi "https://doi.org/10.1007/s00020-022-02685-5" @default.
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