symmetric operators

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1: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
One then needs a self-adjoint extension of a symmetric operator to carry out its spectral theory in a mathematically rigorous manner. An essential feature of such symmetric operators is that their eigenvalues $\lambda$ are real, and eigenfunctions …
We have a direct sum of linear spaces: $\mathcal{D}({T}^{*})=\mathcal{D}(T^{**})+N_{\mathrm{i}}+N_{-\mathrm{i}}$. …
3: 18.38 Mathematical Applications
Define operators $K_{0}$ and $K_{1}$ acting on symmetric Laurent polynomials by $K_{0}=L$ ($L$ given by (18.28.6_2)) and $(K_{1}f)(z)=(z+z^{-1})f(z)$. … The Dunkl type operator is a $q$-difference-reflection operator acting on Laurent polynomials and its eigenfunctions, the nonsymmetric Askey–Wilson polynomials, are linear combinations of the symmetric Laurent polynomial $R_{n}(z;a,b,c,d\,|\,q)$ and the ‘anti-symmetric’ Laurent polynomial $z^{-1}(1-az)(1-bz)R_{n-1}(z;qa,qb,c,d\,|\,q)$, where $R_{n}(z)$ is given in (18.28.1_5). …
4: 1.3 Determinants, Linear Operators, and Spectral Expansions
Real symmetric ($\mathbf{A}=\mathbf{A}^{\mathrm{T}}$) and Hermitian ($\mathbf{A}={\mathbf{A}}^{{\rm H}}$) matrices are self-adjoint operators on $\mathbf{E}_{n}$. …
6: 21.7 Riemann Surfaces
Define the operation
21.7.14 $\boldsymbol{{\eta}}(T_{1}\ominus T_{2})=\boldsymbol{{\eta}}(T_{1})+\boldsymbol% {{\eta}}(T_{2}),$
21.7.15 $4\boldsymbol{{\eta}}^{1}(T)\cdot\boldsymbol{{\eta}}^{2}(T)=\tfrac{1}{2}\left(|% T\ominus U|-g-1\right)\pmod{2},$
7: 2.6 Distributional Methods
2.6.32 $\int_{0}^{\infty}\frac{f(t)}{(t+z)^{\rho}}\,\mathrm{d}t,$ $\rho>0$,
An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). …
8: 32.2 Differential Equations
32.2.13 $z(1-z)I\left(\int_{\infty}^{w}\frac{\,\mathrm{d}t}{\sqrt{t(t-1)(t-z)}}\right)=% \sqrt{w(w-1)(w-z)}\*\left(\alpha+\frac{\beta z}{w^{2}}+\frac{\gamma(z-1)}{(w-1% )^{2}}+(\delta-\tfrac{1}{2})\frac{z(z-1)}{(w-z)^{2}}\right),$
where
9: Bibliography G
• I. M. Gel’fand and G. E. Shilov (1964) Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.
• V. X. Genest, L. Vinet, and A. Zhedanov (2016) The non-symmetric Wilson polynomials are the Bannai-Ito polynomials. Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.
• J. S. Geronimo, O. Bruno, and W. Van Assche (2004) WKB and turning point theory for second-order difference equations. In Spectral Methods for Operators of Mathematical Physics, Oper. Theory Adv. Appl., Vol. 154, pp. 101–138.
• 10: 3.1 Arithmetics and Error Measures
Rounding
Symmetric rounding or rounding to nearest of $x$ gives $x_{-}$ or $x_{+}$, whichever is nearer to $x$, with maximum relative error equal to the machine precision $\frac{1}{2}\epsilon_{M}=2^{-p}$. … The elementary arithmetical operations on intervals are defined as follows: …