# 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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### Proving that the point spectrum of $T$ is not empty

This problem comes from Rudin's Functional Analysis, Chapter 10. It's the problem 15. Suppose $X$ is a Banach space, $T\in\mathcal{B}(X)$ is compact, and $\|T^n\|\geq 1$ for all $n\geq 1$. ...
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### semifinite von Neumann algebra, spectral projection, trace

Let $\mathcal{M}$ be a semifinite von Neumann algebra and $\tau$ be a semifinite faithful normal trace on it. Let $T,P_1,P_2\in \mathcal{M}$, where $P_1,P_2$ are projections with $P_1\perp P_2$. Then, ...
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### Does a polynomial function on spectrum uniquely define polynomial on operator?

Let $X\subset\mathbb C$ be a compact set, let $T$ be a bounded operator with its spectrum contained in $X$, let $P$ be a polynomial. Is it true that whenever $P=0$ on $X$ then $P(T)=0$?
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### Reference request: monotone and strongly monotone with respect to derivatives

Recall, let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be monotone if $$\langle F(u)-F(v), u-v\rangle\geq 0, \quad \forall u,v\in H;$$ strongly monotone if there exists ...
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### When can I Taylor expand a function of an operator?

1-) Is the expression $f(A) = \sum_n \frac{f'(0)}{n!}(A)^n$ always meaningful for any diagonalizable linear operator $A$ and for any analytic function $f$? This seems strange to me because then I ...
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### Is the set of adjoint operators weak* closed?

Suppose we have a Banach space $X$ and a net of bounded operators $(T_\gamma)$ on $X$ such that $T_\gamma^*\to S$, for some bounded operator $S$ on $X^*$, where the convergence is with respect to the ...
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### Construction of Sobolev space

I am reading about the construction of Sobolev spaces from $L^2$. In the book I read (Introduction to Partial Differential Equations, by Gerald B. Folland) that the operator used to construct those ...
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### Are there identities which will make calculating $\Delta \frac{e^{ik|x|}}{|x|}$ more efficient and clearer?

Let $x = (x_1, x_2, x_3)$ be a point in $\mathbb{R}^3$. I am trying to calculate $$\Delta \frac{e^{ik|x|}}{|x|}$$ Doing this 'manually' be calculating second derivatives is really long and tedious ...
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### Spectrum of a positive operator

We know that if $A$ is a self-adjoint unbounded operator on a Hilbert space $(H;\left<.,.\right>)$ then $\sigma(A) \subset \mathbb R$. Now, how it can be shown that if $A$ is more positive i.e. ...
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### Spectrum of an Operator on a Banachspace

Claim: Let $A$ be a bounded linear operator on a Banachspace $\mathfrak{X}$. Denote $\sigma(A)$ as the spectrum of A. Let $\lambda$ be a point in the boundary of the $\sigma(A)$. Then there exist a ...
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### Properness of isometric actions of discrete groups on affine Hilbert spaces

I've been reading Valette's introduction to the Baum-Connes conjecture and as I read the example of a construction of a (model of the) classifying space for proper actions of $\Gamma$ (discrete) given ...
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### Proof that a linear operator is continuous.

Could somebody please verify the following proof I have attempted? It seems too simple so I am worried I have done something wrong.. Many thanks Let $T:(X,\|.\|_X)\to (Y,\|.\|_Y)$ be a linear map ...
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I was recently met with this in my functional analysis class on which I am stuck: Let $\mathbb{H}$ be a Hilbert space and let T be a contraction operator on $\mathbb{H}$ (meaning $||T|| \... 1answer 33 views ### Index of a derivative operator on a circle Let$D: C^{1}(S^{1}) \rightarrow C(S^{1})$be an operator defined as$D(f)=f'$. I would like to find its index (on the road proving that it's a Fredholm operator). First, if$f \in ker(D)$, then$f$... 0answers 43 views ### Two sequences of operators on Hilbert Space Let$H$is some Hilbert Space, and$a_n,b_n \in B(H)$is sequences of some operators on it. We know, that$a_n b_n$converges to$v$by norm. We also know, that all$b_n a_n$are strictly positive. ... 0answers 35 views ### For$\{T_n\}$and$T$positive and self-adjoint, show$T_n \stackrel{SR}{\to} T$iff$(T_n + I)^{-1} \stackrel {s}{\to} (T + I)^{-1}$For$\{T_n\}$and$T$positive and self-adjoint, show$T_n \stackrel{SR}{\to} T$(i.e.$T_n \to T$in the strong resolvent sense) iff$(T_n + I)^{-1} \stackrel {s}{\to} (T + I)^{-1}$(i.e.$(T_n + I)^{...
Let's consider an operator $D: C^{m+n}[a, b] \rightarrow C^{m}[a, b]$, defined as $D(y(t)) = y^{(n)}+a_{n-1}y^{(n-1)}+\ldots+a_{1}y'+a_{0}$, $a_{k} \in C^{m}[a, b]$. I would like to prove that it is ...