![]() ![]() Henstock, The General Theory of Integration (Clarendon Press, Oxford, 1991)ī. 31 (American Mathematical Society, Providence, RI, 1957) American Mathematical Society Colloquium Publications, vol. ![]() Phillips, Functional Analysis and Semigroups. Horn, On the singular values of a product of completely continuous operators. Goldstein, Semigroups of Linear Operators and Applications (Oxford University Press, New York, 1985) Grafakos, Classical and Modern Fourier Analysis (Pearson Prentice-Hall, New Jersey, 2004) Grothendieck, Products tensoriels topologiques et espaces nucleaires. Nagel, et al., One-Parameter Semigroups for Linear Evolution Equations. Schwartz, Linear Operators Part I: General Theory, Wiley Classics edn. Graduate Texts in Mathematics (Springer, New York, 1984) Diestel, Sequences and Series in Banach Spaces. Foiaş, Theory of Generalized Spectral Operators (Gordon Breach, London, 1968)Į.B. Springer Monographs in Mathematics (Springer, New York, 2010) Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces. This process is experimental and the keywords may be updated as the learning algorithm improves. These keywords were added by machine and not by the authors. In the Appendix (Sect. 5.3), we show that all of the results in Chap. 4 have natural analogues for uniformly convex Banach spaces. The results are complete for uniformly convex Banach spaces, so we restrict our presentation to that case, with one exception. We will exploit this new relationship to transfer the theory of semigroups of operators developed for Hilbert spaces to 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. In order to extend the theory to other areas of interest, we begin with a new approach to operator theory on Banach spaces. In this chapter we develop the theory of semigroups of operators, which is the central tool for both. You can read more about these ideas in Schechter's Handbook of Analysis and its Foundations.The Feynman operator calculus and the Feynman path integral develop naturally on Hilbert space. Put another way, you can't even prove $A'$ exists without using AC in an essential way, so you certainly can't construct it concretely. So in such models, $A$ won't have any extension to all of $H$. But from BP you can prove that every everywhere defined operator on any Banach space is bounded. there are models of set theory in which DC and BP both hold (but full AC necessarily fails). ![]() There's a famous theorem of Solovay (extended by Shelah) that it's consistent with DC that every set of reals has the property of Baire (BP) i.e. A common working definition of "concrete" is "something whose existence you can prove using only the axiom of dependent choice (DC)". There's a strong sense in which this is true. Since you used Zorn's lemma in an essential way, you won't get a "concrete" description of such an $A'$. Another way to see this that $A'$ cannot be self-adjoint is to note that, by the closed graph theorem, $A'$ cannot be closed. (In fact you can produce many such operators the extension is highly non-unique.) By Hellinger–Toeplitz, $A'$ cannot be symmetric. Given an unbounded self-adjoint operator $A$ with domain $D(A) \subset H$, using Zorn's lemma you can produce an everywhere defined operator $A'$ on $H$ which extends $A$. Related question: Invertible unbounded linear maps defined on a Hilbert space Now the question is: Since the extension of a linear, unbounded operator to the whole of the space through the AC, will produce a -still- unbounded, linear operator, does the previous remark imply that the extension of linear, self-adjoint, unbounded operators on the whole of the space, produces non-self-adjoint operators? What would be a concrete relevant example? On the other hand, it is well known that any linear map from a subspace of a Banach space $X$ to another Banach space $Y$ can be extended to a linear map $X\to Y$ defined on the whole of $X$ using Zorn's Lemma (see for example: Unbounded linear operator defined on $l^2$). Since the operators of interest in physics are self-adjoint (and thus symmetric) they fall into this. (see also: Riesz-Nagy, "Functional Analysis", 1955, p.296 and also Reed-Simon, "Methods of Modern Mathematical Physics", 1975, p.84). This is a direct consequence of the Hellinger-Toeplitz theorem. I am not quite sure whether this is research-level, but let me state some context first:Īn old result of functional analysis tells us that a symmetric (in the sense that $(Ax,y)=(x,Ay)$, for all $x,y \in H$), unbounded operator $A$, acting on a Hilbert space $H$, cannot be defined on the whole space but only in a dense subspace of it. I am not an expert in functional analysis but I was studying some, motivated from some mathematical physics considerations. ![]()
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