von Neumann’s inequality for commuting, diagonalizable contractions. I
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- by B. A. Lotto
- Proc. Amer. Math. Soc. 120 (1994), 889-895
- DOI: https://doi.org/10.1090/S0002-9939-1994-1169881-8
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Abstract:
We obtain a sufficient condition for an $n$-tuple $T$ of commuting, diagonalizable contractions on a finite-dimensional space to satisfy von Neumann’s inequality $||p(T)|| \leqslant ||p|{|_\infty }$ for any polynomial $p$ in $n$ variables, where $||p|{|_\infty }$ denotes the supremum of $|p|$ over the unit polydisk in ${{\mathbf {C}}^n}$. We apply this condition to the case where $T$ acts on a two- or three-dimensional space. In addition, we prove that von Neumann’s inequality for commuting, diagonalizable contractions on a three-dimensional space implies von Neumann’s inequality for arbitrary commuting contractions on a three-dimensional space.References
- T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88–90. MR 155193
- M. J. Crabb and A. M. Davie, von Neumann’s inequality for Hilbert space operators, Bull. London Math. Soc. 7 (1975), 49–50. MR 365179, DOI 10.1112/blms/7.1.49
- S. W. Drury, Remarks on von Neumann’s inequality, Banach spaces, harmonic analysis, and probability theory (Storrs, Conn., 1980/1981) Lecture Notes in Math., vol. 995, Springer, Berlin, 1983, pp. 14–32. MR 717226, DOI 10.1007/BFb0061886
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- John A. R. Holbrook, Polynomials in a matrix and its commutant, Linear Algebra Appl. 48 (1982), 293–301. MR 683226, DOI 10.1016/0024-3795(82)90115-X
- Keith Lewis and John Wermer, On the theorems of Pick and von Neumann, Function spaces (Edwardsville, IL, 1990) Lecture Notes in Pure and Appl. Math., vol. 136, Dekker, New York, 1992, pp. 273–280. MR 1152352
- B. A. Lotto and T. Steger, von Neumann’s inequality for commuting, diagonalizable contractions. II, Proc. Amer. Math. Soc. 120 (1994), no. 3, 897–901. MR 1169882, DOI 10.1090/S0002-9939-1994-1169882-X
- Donald Sarason, Generalized interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 127 (1967), 179–203. MR 208383, DOI 10.1090/S0002-9947-1967-0208383-8
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
- N. Th. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Functional Analysis 16 (1974), 83–100. MR 0355642, DOI 10.1016/0022-1236(74)90071-8
- Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124
Bibliographic Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 889-895
- MSC: Primary 47A30; Secondary 15A60, 47B99
- DOI: https://doi.org/10.1090/S0002-9939-1994-1169881-8
- MathSciNet review: 1169881