Extended tensor products and an operator-valued spectral mapping theorem

V. G. Kurbatov ; I. V. Kurbatova

Izvestiya. Mathematics , Tome 79 (2015) no. 4, pp. 710-739.

Résumé

We introduce the notion of an extended tensor product of Banach spaces $X$ and $Y$. It is defined as a triple consisting of a Banach space $X\boxtimes Y$ and two full subalgebras $\mathbf B_0(X)$ and $\mathbf B_0(Y)$ of the algebras $\mathbf B(X)$ and $\mathbf B(Y)$ of all bounded linear operators on $X$ and $Y$ respectively. It is assumed that $X\boxtimes Y$ is an extension of the ordinary tensor product $X\otimes Y$, and the functionals on $X^*\otimes Y^*$ and operators on $\mathbf B_0(X)\otimes\mathbf B_0(Y)$ have a canonical extension from $X\otimes Y$ to $X\boxtimes Y$. Every pseudo-resolvent $\mathbf B_0(Y)$ generates a functional calculus that sends analytic $\mathbf B_0(X)$-valued functions in a neighbourhood of the singular set of the pseudo-resolvent to operators on $X\boxtimes Y$. We prove an analogue of the spectral mapping theorem for such a functional calculus.

Détail

Keywords: tensor product, pseudo-resolvent, spectral mapping theorem.

Bibliographie

[1] A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc., 1955, no. 16, 1955, 140 pp. | MR

[2] R. Schatten, A theory of cross-spaces, Ann. of Math. Stud., 26, Princeton Univ. Press, Princeton, NJ, 1950, vii+153 pp. | MR | Zbl

[3] G. Köthe, Topological vector spaces. I, Grundlehren Math. Wiss., 159, Springer-Verlag, New York, 1969, xv+456 pp. | MR | Zbl

[4] G. Köthe, Topological vector spaces. II, Grundlehren Math. Wiss., 237, Springer-Verlag, New York, 1979, xii+331 pp. | DOI | MR | Zbl

[5] A. Ya. Helemskii, Lectures and exercises on functional analysis, Transl. Math. Monogr., 233, Amer. Math. Soc., Providence, RI, 2006, xviii+468 pp. | MR | Zbl

[6] M. R. Embry, M. Rosenblum, “Spectra, tensor products, and linear operator equations”, Pacific J. Math., 53:1 (1974), 95–107 | DOI | MR | Zbl

[7] V. G. Kurbatov, “The spectrum of an operator with commensurable deviations of the argument and with constant coefficients”, Differential Equations, 13:10 (1977), 1231–1235 | MR | Zbl

[8] V. G. Kurbatov, I. S. Frolov, “An operator variant of the maximum principle”, Math. Notes, 36:4 (1984), 763–765 | DOI | MR | Zbl

[9] V. G. Kurbatov, Functional differential operators and equations, Math. Appl., 473, Kluwer Acad. Publ., Dordrecht, 1999, xx+433 pp. | MR | Zbl

[10] M. I. Gil', “Invertibility and spectra of operators in tensor products of Hilbert spaces”, Glasg. Math. J., 46:1 (2004), 101–116 | DOI | MR | Zbl

[11] M. I. Gil', “Estimates for resolvents and functions of operator pencils on tensor products of Hilbert spaces”, Kyoto J. Math., 51:3 (2011), 673–686 | DOI | MR | Zbl

[12] E. Hille, R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., 31, rev. ed., Amer. Math. Soc., Providence, RI, 1957, xii+808 pp. | MR | MR | Zbl

[13] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag, New York, 1966, xix+592 pp. | MR | MR | Zbl | Zbl

[14] W. Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York–Düsseldorf–Johannesburg, 1973, xiii+397 pp. | MR | MR | Zbl

[15] A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Transl. Math. Monogr., 71, Amer. Math. Soc., Providence, RI, 1988, iv+250 pp. | MR | MR | Zbl | Zbl

[16] Kh. D. Ikramov, “Matrix pencils: Theory, applications, and numerical methods”, J. Soviet Math., 64:2 (1993), 783–853 | DOI | MR | Zbl

[17] I. V. Kurbatova, “Banach algebras associated with linear operator pencils”, Math. Notes, 86:3 (2009), 361–367 | DOI | DOI | MR | Zbl

[18] N. Bourbaki, Éléments de mathématique, Fasc. XXXII. Théories spectrales. Chap. I: Algèbres normées. Chap. II: Groupes localement compacts commutatifs, Actualités Sci. Indust., 1332, Hermann, Paris, 1967, iv+166 pp. | MR | MR | Zbl | Zbl

[19] B. Simon, “Uniform crossnorms”, Pacific J. Math., 46:2 (1973), 555–560 | DOI | MR | Zbl

[20] R. Arens, “The adjoint of a bilinear form”, Proc. Amer. Math. Soc., 2:6 (1951), 839–848 | DOI | MR | Zbl

[21] B. V. Shabat, Vvedenie v kompleksnyi analiz, Nauka, M., 1969, 576 pp. | MR | Zbl

[22] N. Bourbaki, Éléments de mathématique, Fasc. X. Première partie. Livre III: Topologie générale. Chapitre 10: Espaces fonctionnels, Actualités Sci. Indust., 1084, Hermann, Paris, 1961, 96 pp. | MR | MR | Zbl | Zbl

[23] H. H. Schaefer, Topological vector spaces, The Macmillan Co., New York; Collier–Macmillan Ltd., London, 1966, ix+294 pp. | MR | MR | Zbl | Zbl

[24] A. G. Baskakov, K. I. Chernyshov, “Spectral analysis of linear relations and degenerate operator semigroups”, Sb. Math., 193:11 (2002), 1573–1610 | DOI | DOI | MR | Zbl

[25] F. Räbiger, M. P. H. Wolff, “Spectral and asymptotic properties of resolvent-dominated operators”, Tübinger Berichte zur Funktionalanalysis, 7:1 (1997/1998), 217–235

[26] M. S. Bichegkuev, “Conditions for solubility of difference inclusions”, Izv. Math., 72:4 (2008), 647–658 | DOI | DOI | MR | Zbl

[27] I. V. Kurbatova, “Nekotorye svoistva lineinykh otnoshenii”, Vestn. f-ta PMM, no. 7, VGU, Voronezh, 2009, 68–89

[28] L. D. Kudryavtsev, Kratkii kurs matematicheskogo analiza, Nauka, M., 1989, 736 pp. | Zbl

[29] H. A. Gindler, A. E. Taylor, “The minimum modulus of a linear operator and its use in spectral theory”, Studia Math., 22 (1962/63), 15–41 | MR | Zbl

[30] A. Pietsch, Operator ideals, Math. Monogr., 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978, 451 pp. | MR | Zbl | Zbl

[31] L. W. Weis, “On the computation of some quantities in the theory of Fredholm operators”, Rend. Circ. Mat. Palermo (2), 1984, suppl., no. 5, 109–133 | MR | Zbl

[32] A. G. Baskakov, “Representation theory for Banach algebras, Abelian groups, and semigroups in the spectral analysis of linear operators”, J. Math. Sci. (N. Y.), 137:4 (2006), 4885–5036 | DOI | MR | Zbl

[33] Kh. D. Ikramov, Chislennoe reshenie matrichnykh uravnenii, Nauka, M., 1984, 192 pp. | MR | Zbl