Abstract
The Feynman operator calculus and the Feynman path integral develop naturally on Hilbert space. In this chapter we develop the theory of semigroups of operators, which is the central tool for both. In order to extend the theory to other areas of interest, we begin with a new approach to operator theory on Banach spaces. We first show that the structure of the bounded linear operators on Banach space with an S-basis is much closer to that for the same operators on Hilbert space. We will exploit this new relationship to transfer the theory of semigroups of operators developed for Hilbert spaces to Banach spaces. The results are complete for uniformly convex Banach spaces, so we restrict our presentation to that case, with one exception. In the Appendix (Sect. 5.3), we show that all of the results in Chap. 4 have natural analogues for uniformly convex Banach spaces.
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References
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics (Springer, New York, 2010)
I. Colojoar\(\breve{\mathrm{a}}\), C. Foiaş, Theory of Generalized Spectral Operators (Gordon Breach, London, 1968)
E.B. Davies, The Functional Calculus. J. Lond. Math. Soc. 52, 166–176 (1995)
J. Diestel, Sequences and Series in Banach Spaces. Graduate Texts in Mathematics (Springer, New York, 1984)
N. Dunford, J.T. Schwartz, Linear Operators Part I: General Theory, Wiley Classics edn. (Wiley Interscience, New York, 1988)
K.J. Engel, R. Nagel, et al., One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194 (Springer, New York, 2000)
A. Grothendieck, Products tensoriels topologiques et espaces nucleaires. Mem. Am. Math. Soc. 16, 1–140 (1955)
L. Grafakos, Classical and Modern Fourier Analysis (Pearson Prentice-Hall, New Jersey, 2004)
J.A. Goldstein, Semigroups of Linear Operators and Applications (Oxford University Press, New York, 1985)
H.H. Kuo, Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics, vol. 463 (Springer, New York, 1975)
A. Horn, On the singular values of a product of completely continuous operators. Proc. Natl. Acad. Sci. 36, 374–375 (1950)
E. Hille, R.S. Phillips, Functional Analysis and Semigroups. American Mathematical Society Colloquium Publications, vol. 31 (American Mathematical Society, Providence, RI, 1957)
R. Henstock, The General Theory of Integration (Clarendon Press, Oxford, 1991)
B. Helffer, J. Sjöstrand, in Équation de Schrödinger avec champ magnetique et équation de Harper, Schrödinger Operators (Snderborg, 2988), ed. by H. Holden, A. Jensen. Lecture Notes in Physics, vol. 345 (Springer, Berlin, 1989), pp. 118–197
W.B. Johnson, H. Konig, B. Maurey, J.R. Retherford, Eigenvalues of p-summing and l p type operators in Banach space. J. Funct. Anal. 32, 353–380 (1978)
T. Kato, Perturbation Theory for Linear Operators, 2nd edn. (Springer, New York, 1976)
T. Kato, Trotters product formula for an arbitrary pair of selfadjoint contraction semigroups, in Advances in Mathematics: Supplementary Studies, vol. 3 (Academic, New York, 1978), pp. 185–195
S. Kakutani, On equivalence of infinite product measures. Ann. Math. 49, 214–224 (1948)
T. Lalesco, Une theoreme sur les noyaux composes. Bull. Acad. Sci. 3, 271–272 (1914/1915)
V.B. Lidskii, Non-self adjoint operators with a trace. Dokl. Akad. Nauk. SSSR 125, 485–487 (1959)
G. Lumer, R.S. Phillips, Dissipative operators in a Banach space. Pacific J. Math. 11, 679–698 (1961)
G. Lumer, Spectral operators, Hermitian operators and bounded groups. Acta. Sci. Math. (Szeged) 25, 75–85 (1964)
A. Pietsch, History of Banach Spaces and Operator Theory (Birkhäuser, Boston, 2007)
A. Pietsch, Einige neue Klassen von kompacter linear Abbildungen. Revue der Math. Pures et Appl. (Bucharest) 8, 423–447 (1963)
A. Pietsch, Eigenvalues and s-Numbers (Cambridge University Press, Cambridge, 1987)
T.W. Palmer, Unbounded normal operators on Banach spaces. Trans. Am. Math. Sci. 133, 385–414 (1968)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44 (Springer, New York, 1983)
J.R. Retherford, Applications of Banach ideals of operators. Bull. Am. Math. Soc. 81, 978–1012 (1975)
B. Simon, Trace Ideals and Their Applications. London Mathematical Society Lecture Notes Series, vol. 35 (Cambridge University Press, New York, 1979)
B. Simon, Notes on infinite determinants of Hilbert space operators. Adv. Math. 24, 244–273 (1977)
E.M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton University Press, Princeton, 1970)
V.A. Strauss, C. Trunk, Spectralizable operators. Integr. Equ. Oper. Theory 61, 413–422 (2008)
V.A. Vinokurov, Yu. Petunin, A.N. Pliczko, Measurability and regularizability mappings inverse to continuous linear operators (in Russian). Mat. Zametki. 26(4), 583–591 (1979). English translation: Math. Notes 26, 781–785 (1980)
I. Vrabie, C 0 -Semigroups and Applications. North-Holland Mathematics Studies, vol. 191 (Elsevier, New York, 2002)
H. Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation. Proc. Natl. Acad. Sci. 35, 408–411 (1949)
K. Yosida, Functional Analysis, 2nd edn. (Springer, New York, 1968)
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Gill, T.L., Zachary, W. (2016). Operators on Banach Space. In: Functional Analysis and the Feynman Operator Calculus. Springer, Cham. https://doi.org/10.1007/978-3-319-27595-6_5
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