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Linear Algebra for Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
Hermitian operator or self-adjoint operator in matrix algebra is called Hermitian matrix, A, when it is equal to its conjugate transpose, AH or A† (A dagger). A=AH=A†,AisHermetianAH=A†=(A*)T=(AT)*
Operator Theory
Published in Hugo D. Junghenn, Principles of Analysis, 2018
For any S∈B(H) $ S\in \mathcal{B}(\mathcal{H}) $ , the operators S∗S $ S^*S $ , SS∗ $ SS^* $ , S+S∗ $ S+S^* $ and i(S-S∗) $ i(S-S^*) $ are self-adjoint. These examples suggests that self-adjoint operators may be viewed as the analogs of real numbers in the complex number system, the adjoint operation being the analog of conjugation. The following proposition strengthens this analogy. The proof is left as an exercise for the reader (12.2).
Hilbert Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Let T be a compact operator from a Hilbert space U into a Hilbert space V. Show that:T∗T $ T^{*}T $ is a compact, self-adjoint, positive semi-definite operator from a space U into itself.All eigenvalues of a self-adjoint operator on a Hilbert space are real.Conclude that all eigenvalues of T∗T $ T^{*}T $ are real and nonnegative.
On the Sturm–Liouville problem describing an ocean waveguide covered by pack ice
Published in Applicable Analysis, 2022
Boris P. Belinskiy, Don B. Hinton, Lakmali Weerasena, Mohammad M. Khan
We finally note that the modern literature contains study of diverse wave propagation problems in the ocean waveguides, with ice-covered or partially ice-covered surface, including waveguides of varying depth (see, e.g. [2, 5–9, 11, 17, 18, 31]). Further, our study is based on the theory of self-adjoint operators. Yet, the similar study of viscoelastic-type models (see, e.g. [36]) would require using more complex methods of non-self-adjoint operators. We believe that further study of wave propagation in the layered ocean covered by ice may require a spectral analysis similar to the analysis in our paper. The question appears whether Propositions 3.3, 3.5, 3.6, Remarks 3.4, 4.5, 5.4, 5.5, 5.6, Theorems 4.2, 4.3,5.1, numerical analysis in Section 6, and asymptotic analysis in Section 7 hold (or may be generalized) in the case of the boundary condition more complex than (12), e.g. (80). We hope to study these issues later.
Weak, strong and linear convergence of the CQ-method via the regularity of Landweber operators
Published in Optimization, 2020
Andrzej Cegielski, Simeon Reich, Rafał Zalas
The proof follows from the basic properties of compact self-adjoint operators and the spectral decomposition theorem applied to ; see, for example [50, Section 15.3] or [51, Section 4.8]. We remark in passing that the spectral decomposition theorem is usually presented in the setting of a complex Hilbert space. Nevertheless, one can obtain an analogous result in a real Hilbert space by using, for instance, a complexification argument combined with the fact that all eigenvalues of a self-adjoint operator are real. Hence, when referring to results from [50, Section 15.3] we actually adjust them to the setting of a real Hilbert space.