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 *A*^{2} 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}^{∞}
*λ*_{i}
*x*_{i}(*x*)
*y*_{i}, with
|| *x*_{i} || ≤ 1,
|| *y*_{i} || ≤ 1,
and {*λ*_{i}}
in *l*^{1};
One can then define the trace of *T* by tr *T*:=
inf ∑_{i=1}^{∞}
*λ*_{i} .

@ __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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