Operator Theory| 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 : X → Y 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 72–78;
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 A† ⊃ A), 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 : X
→ Y 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];
Cintio & Michelangeli a2012 [and hermitian, as physical observables].
@ 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].
@ Jordan operator algebras: Blecher & Neal a1709 [as general setting for non-commutative topology];
Blecher & Wang a1812 [theory].
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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