# 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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### Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
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### Perturbation of operators and eigenvalues

Suppose $P\in\mathcal{B}(\mathcal{H})$ is a self-adjoint compact operator. Lets perturb $P$ by multiplying it by a bounded operator $S$ and set $T=PS.$ Then what can be said for the spectrum of $T?$ ...
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### functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
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### Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
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### Strong closure of a C*-algebra of operators.

In Arveson's book, the Kaplansky density theorem is proved in order to have this corollary: "Let $A$ be a self-adjoint algebra of operators on a separable Hilbert space $H$. Then for every operator ...
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### Prove that $\frac{1}{\phi(D)}c=c\frac{1}{\phi(0)}$

Could someone prove the following? $$\frac{1}{\phi(D)}c=c\frac{1}{\phi(0)}$$ where $D$ is ${\frac{d}{dx}}$ and $c$ is a constant. for example $$\frac{1}{D^4+2D+3}c=c\frac{1}{0+0+3}=\frac{c}{3}$$
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### Invariant subspaces and their complements

The following is Exercise 6.4.2 of Conway's Functional Analysis: Suppose $T\in B(X)$ where $X$ is a Banach space. Prove that $M$ is invariant under $T$ if and only if $M^{\perp}$ is invariant under ...
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### How find two differential operator $A$ and $B$ such $A\circ \dfrac{d}{dx}=B\circ x$

Question: show that there exists differential operators $A$ and $B$ where $$A=\sum_{k=0}^{n}a_{k}(x)\dfrac{d^k}{dx^k}\neq 0,B=\sum_{k=0}^{n}b_{k}(x)\dfrac{d^k}{dx^k}\neq 0$$ ...
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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 ...
### 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 ...