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Slodkowski joint spectrum and tensor product. (English) Zbl 1017.47500

In an already classical paper, J. L. Taylor [J. Funct. Anal. 6, 172-191 (1970; Zbl 0233.47024)] introduced a joint spectrum for tuples of commuting Banach space operators via the associated Koszul complex. In the case of Hilbert space operators, the joint spectrum can be characterized in terms of invertibility, as first noticed by the reviewer [F.-H. Vasilescu, Rev. Roum. Math. Pures Appl. 22, 1003-1009 (1977; Zbl 0371.47035)]. Using the partial non-exactness in the Koszul complex, Z. Słodkowski [Stud. Math. 61, 239-255 (1977; Zbl 0369.47021)] introduced some partial joint spectra for commuting tuples of Banach space operators. In the present paper, the authors characterize the joint spectra introduced by Słodkowski in the context of Hilbert spaces, following some ideas of R. E. Curto [Trans. Am. Math. Soc. 266, 129-159 (1981; Zbl 0457.47017)], and apply their results to obtain formulas concerning these spectra for some tuples obtained via tensor products.

MSC:

47A13 Several-variable operator theory (spectral, Fredholm, etc.)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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