Abstract
Given several bounded linear operators \(A_1,..., A_n\) on a Hilbert space, their projective spectrum is the set of complex vectors \(z=(z_1,..., z_n)\) such that the multiparameter pencil \(A(z)=z_1A_1+\cdots +z_nA_n\) is not invertible. This paper studies the projective spectrum of the shift operator T, its adjoint \(T^*\) and a projection operator P. Two spaces of concern are the classical Bergman space \(L_a^2(\mathbb {D})\) and the \(L^2\) space over the torus \({\mathbb T}^2\). The projective spectra are completely determined in both cases. The results lead to new questions about Toeplitz operators.
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Communicated by Mihai Putinar.
In Memory of Jörg Eschmeier.
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The first-named author is supported by China Scholarship Council (201908210167). This research is supported by NNSF of China (11501277).
This article is part of the topical collection “Multivariable Operator Theory. The Jörg Eschmeier Memorial” edited by Raul Curto, Michael Hartz and Mihai Putinar.
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Cui, P., Yang, R. A Note on Joint Spectrum in Function Spaces. Complex Anal. Oper. Theory 17, 81 (2023). https://doi.org/10.1007/s11785-023-01383-3
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DOI: https://doi.org/10.1007/s11785-023-01383-3