# Tagged Questions

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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### Give necessary and sufficient conditions for a multiplication on $L^p$ to be compact

Let $(X, \Omega, \mu)$ be a $\sigma-$ finite measure space and for $\phi \in L^\infty(\mu)$ let $M_\phi:L^p(\mu) \to L^p(\mu)$ defined by $M_\phi f = \phi f$ be the multiplication operator. Give ...
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### If $T$ is self-adjoint, is the set of power series in $T$ closed?

If $T$ is a bounded self-adjoint operator on a Hilbert space, is the set of convergent power series in $T$ closed in the norm topology? I ask because I'm reading some spectral theorems and I was ...
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### Is an Invariant set Connected?

Let an autonomous dynamical system is characterized by the state equation $$\dot x(t) = f(x(t)),\quad x(0)=x_0$$ with state $x(t)\in \mathbb R^n$. The definition of invariant set, as I came across, ...
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### Solution set of linear operator equations

Suppose $\mathcal{X}$ and $\mathcal{Y}$ are two Hilbert spaces. Let $A:\mathcal{X} \mapsto \mathcal{Y}$ be a bounded linear operator. Consider a linear operator equation $Ax=b$. My question is what ...
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### $\sup$ norm of a function

The following is an example of Murphy's C*-algebras and operator theory: I do not know how he concludes $$\int_0^1 |k(s,t) - k(s',t)||f(t)| dt \leq \sup|k(s,t) - k(s',t)|||f||_\infty$$ Please help ...
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### Periodic Laplace operator non closed in $C^2(0,L)$

How can I show that the Laplacian operator is not closed in the domain $D=\{f \in C^2(0,L) \mid \mbox{ f is vanishing in a neighborhood of 0 and L } \}$ for a fixed $L$? And how can I show that it is ...
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### characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...