Questions tagged [operator-theory]

Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.

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Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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Numerical range of the first derivative operator on $\{ u \in H^1(0,1): u(1)=0 \}$

I need to calculate the numerical range of the operator $T:D(T)\subseteq L^2(0,1) \to L^2(0,1)$ defined by $$D(T):=\{ u \in H^1(0,1): u(1)=0 \}, \ Tu:=u', \ u \in D(T),$$ where $H^1(0,1)$ is the ...
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Using abstract Hilbert spaces to solve differential equations

There are techniques for solving PDE's, such as Fock-Schwinger method in physics, which involve translating the problem from the language of distributions to the language of the abstract Hilbert ...
If $E$ is a $\mathbb R$-Banach space, $(T(t))_{t\ge0}$ is a semigroup of bounded linear operators on $E$ and $(\mathcal D(A),A)$ denotes the generator of $(T(t))_{t\ge0}$, is $(\mathcal D(A),A)$ ...