Operator Theory  

In General > s.a. measure [operator-valued].
* History: Operator theory was inspired and motivated by the development of quantum physics.
$ Def: An operator is a linear map L : XY between Banach spaces, or Hilbert spaces.
* Operations on operators: Adjoint, extensions (e.g., Friedrich extension).
@ General references: Murray & Von Neumann AM(36), AM(43); Dunford & Schwartz 58, 63, 71; Dixmier 69; Beals 71; Zhu 07 [on function spaces].
@ Hilbert space: Achiezer & Glazman 61; Cirelli & Gallone 74; Reed & Simon 7278; Schechter 81; Lundsgaard Hansen 16.
@ Related topics: Atiyah 74 [elliptic]; Lahti et al JMP(99) [operator integrals].

Types of Operators > s.a. Projector; Subnormal; Symmetric.
* Bounded: An operator A: \(\cal H\)1 → \(\cal H\)2 such that there exists c in \(\mathbb R\) such that, for all v in \(\cal H\)1, || Av ||2 < c || v ||1 (c is its norm).
* Hermitian: An operator A on a Hilbert space such that \(\langle\)φ, Aψ\(\rangle\) = \(\langle\)Aφ, ψ\(\rangle\), for all φ and ψ in its domain \(\cal D\)(A), i.e., complex symmetric; Relationships: This condition is weaker than self-adjointness (it could be that AA), but it still implies that \(\langle\)ψ, Aψ\(\rangle\) ∈ \(\mathbb R\); However, if A and A2 are maximal hermitian, then A is self-adjoint.
* Essentially self-adjoint: A Hermitian one defined on a dense subspace; It admits a unique self-adjoint extension.
* Self-adjoint: An operator A such that A* = A, i.e., A is Hermitian and with \(\cal D\)(A*) = \(\cal D\)(A); Useful because (1) The eigenvalues are real; (2) There are complete orthonormal sets of eigenvectors; (3) One can meaningfully define functions f(A), with f Borel measurable.
* Normal: A matrix/operator A such that A*A = AA*; It can be diagonalized; The unitary geometry of the rows is the same as that of the columns.
* Trace-class: An operator T : XY between Banach spaces which can be written as T(x) = ∑i=1 λi xi(x) yi, with || xi || ≤ 1, || yi || ≤ 1, and {λi} in l1; One can then define the trace of T by tr T:= inf ∑i=1 λi .
@ General references: Reed & Simon 72 [infinite-dimensional spaces]; Uhlmann SCpma(16)-a1507 [antilinear operators].
@ Self-adjoint: Dubin et al JPA(02) [spectral and semispectral measures]
@ Self-adjoint extensions: Bonneau et al AJP(01)mar-qp [self-adjoint extension examples]; Ibort & Pérez-Pardo a1502-ln [and physics]; > s.a. scattering.
@ PT-symmetric: Caliceti et al JPA(07)-a0705 [non-selfadjont, with real discrete spectrum].
@ Unbounded: Bagarello RVMP(07), a0903 [algebras, intro and applications]; Jorgensen a0904 [duality theory].

Spaces of Operators / Operator Algebras > s.a. observable algebras [von Neumann].
* \(\cal B\)(\(\cal H\)): The space of bounded operators on a (separable) Hilbert space, a W*-algebra; Its topological dual is the space of trace-class operators.
* \(\cal K\)(\(\cal H\)): The space of compact operators on \(\cal H\); Its topological dual is \(\cal B\)(\(\cal H\)).
@ General references: Li Bing-Ren 92 [intro]; Blackadar 06 [C*-algebras and von Neumann algebras]; Lledó a0901 [operator algebras, informal overview]; Reyes-Lega proc(16)-a1612 [in quantum physics, rev].
@ Maps between operator algebras: Salgado & Sánchez-Gómez mp/04-conf [Jamiolkowski positivity criterion].

Related Concepts > s.a. Boundary-Value Problems; matrix [determinant]; norm; series [Taylor]; Stone's Theorem.
@ Spectral theory: Cirelli 72; Müller-Pfeiffer 81; Friedrichs 80; Weidmann 87; Hislop & Sigal 95; Laugesen a1203-ln [for self-adjoint partial differential operators]; > s.a. Bloch Theory, Toeplitz.
@ Eigenvalues / eigenvectors: Andrew & Miller PLA(03) [continuous + point spectrum, Lanczos algorithm]; Georgescu a0811 [purely discrete].
@ Operations on operators: Gill & Zachary JPA(05) [fractional powers of linear operators]; Babusci & Dattoli a1105 [logarithm].

Differential Operators > s.a. conformal structures [invariant]; Derivatives; D'Alembertian; laplacian.
* Zero modes: The eigenfunctions of a differential operator with zero eigenvalue; They physically correspond to massless excitations of the field.
* Pseudodifferential operators: Used in quantization, especially in the phase space (Wigner-Moyal) formulation and in quantum field theory in curved spacetime.
@ References: Atiyah in(75); Giacomini & Mouchet JPA(07)-a0706 [1D, finding gaps in the spectrum]; Esposito & Napolitano NCC(15)-a1509 [pseudodifferential operators on Riemannian manifolds].

In Physics > s.a. observable algebras.
@ General references: Sakai 91 [dynamical systems]; Balinsky & Evans 10 [relativistic operators].
@ Quantum theory: Jordan 69; De Lange & Raab 91; Svozil qp/96 [discrete operators and observables]; Bonneau et al AJP(01)mar-qp [self-adjoint extensions]; Kempf PRD(01) [symmetric operators and symmetries]; D'Ariano PLA(02) [universal observables]; Ozorio de Almeida & Brodier qp/05/JPA [semi-classical evolution]; de Oliveira 09; Recami et al IJMPA(10) [non-self-adjoint operators]; Ruetsche SHPMP(11) [normal operators]; Moretti 13; > s.a. annihilation and creation operators; formulations and representations of quantum mechanics.
@ Unitary operators: Accardi & Sabbadini qp/00 [enhancing specified components].

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