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Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

0 votes
1 answer
114 views

Minimal norm problem whose unknown is an operator

Generally given an Hilbert space $X$ with and a bounded linear operator $H : X \to X$ given a vector $y \in X$ we seek an $x \in X$ such that $$ f(x) = \frac{1}{2} \left\lVert Hx - y \right\rVert_2^2 …
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2 votes
0 answers
94 views

Self adjoint operators from energy functionals

It is known that the equation $$ \Delta f = 0 $$ on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet Ene …
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-3 votes
1 answer
75 views

Minimal norm problem with linear combination of translation operator to be estimated

Follow up question from this one Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form $$ H = H(\alpha_1 …
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