# 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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### How to find the poles of a green function?

I am trying to construct a green function for $y''+\alpha^2u=f(x), u(0)=u(1), u'(0)=u'(1)$. For that I am trying to follow the procedure described here:(Construct the Green s function for the equation)...
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### Showing that an operator is bijective

Assume that $A$ generates a contraction semigroup on a Hilbert space $X$, and B is a bounded linear operator on $X$. I want to show that $A + B - 2|| B ||I$ with the domain equal to the domain ...
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### An invertible sparse matrix?

I'm not entirely certain about how to tackle this problem.... I hope you ladies and gents can help :) If $M\in M_{n\times n}(\mathbb{R})$ be such that every row has precisely tow non-zero entries, ...
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### Why is the Calkin algebra purely infinite?

I tried using the fact that in a simple unital $C^*$-algebra, $\mathcal{A}$, purely infinite is equivalent to the following: If $x\in\mathcal{A}$ is non-zero, then there exists $a,b\in\mathcal{A}$ ...
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### Prove that norm of Yosida approximations blows up

I need an help with the following exercise about Yosida approximations. Let's fix the notation: let $A$ be an unbounded linear operator define on a dense domain $D(A)\subset X$ of a Banach space $X$. ...
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Suppose $\{u_n\}$ is a convergent sequence in Hilbert space $H$ and $L$ is a bounded (continuous) linear operator on $H$. Use the definition of convergence to show that $\lim_{n\to\infty}(Lu_n) = L(\... 0answers 61 views ### Find eigenfunctions of the integral operator with kernel$\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$Find the eigenvalue and eigenfunctions of the integral operator$Ku=\int_0^\pi k(x,y)u(y)dy$.$k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ... 1answer 45 views ### Is it necessary to use the Hahn-Banach theorem to show that$(X/M)^*\simeq M^\perp$? Let$X$be a Banach space with dual space$X^*$, and let$M$be a closed subspace of$X$. Then$M^\perp=\{x^*\in X^*: x*(m)=0 \text{ for all } m\in M\}$is a closed subspace in$X^*$. I read the ... 3answers 106 views ### looking for help with a trace/norm inequality I'm trying to understand a derivation that seems to claim that$\left\vert\text{Tr}\left[\rho U^\dagger\left[U,O\right]\right]\right\vert\leq\|\left[U,O\right]\|$, where$\rho$is Hermitian and has ... 1answer 76 views ### When can we exchange the trace and an integral/limit/derivative? For a trace class operator$A$(acting on a Hilbert space), that is parameterised by a real variable$x$, what are the conditions for the following to hold? $$\mathrm{tr} \int_a^b A(x) \, dx = \int_a^... 1answer 30 views ### symetric closed operator and extension [closed] i have this question let A a symetric closed operator let pose that A have a self adjoint extension is possible that A has an extension such that closure A can't have a self adjoint extension 1answer 23 views ### What can one assume about T^* when showing that T is normal? Consider a continuous and linear operator T such that$$ T : l^2 \to l^2 $$where (a_n) \mapsto (\alpha_n a_n) Moreover (\alpha_n) is a sequence of complex numbers that converges to zero. Now, ... 0answers 32 views ### Multiplication operators are sectorial Consider the multiplication operator M_a on L^p(M), where M is a Riemannian manifold, and a is a non-negative function. An operator A is said to be sectorial if there exists \theta \in (0, \... 1answer 130 views ### Showing that an operator generates a contraction semigroup Let A be the infinitesimal generator of a contraction semigroup (T(t))_{t\ge 0} on the Hilbert space X, and D\in\mathcal{L}(X). I want to show that the operator A+D-2\|D\|I with domain D(A)... 0answers 18 views ### Reference for measures of commutativity needed I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n \times n Hermitian matrices, and [A,B]=C. I'd like a ... 1answer 93 views ### Why has the Stein operator for normal approximations the form (\mathcal Af)(x)=f^\prime(x)-xf(x)? My Question: Why has the Stein operator \mathcal A for normal approximations the form (\mathcal Af)(x)=f^\prime(x)-xf(x)? How can one deduce this form of the operator? Reason for my question: I ... 2answers 30 views ### A relation between the domain of A and the domain of \bar A Let A be an operator:$$ A:D(A)\to R(A) $$where D(A) and R(A) are respectively the domain and the range of A and they are subspaces of a Hilbert spcae (H,\|\|). Suppose that A is a ... 2answers 75 views ### proving that \text{aff}C-\text{aff}C\subset\text{aff}\,(C-C) In proof of Theorem 6.4.1 of Auslender's book about asymptotic cones, the author assumes that \text{rge}\,A\subset\text{aff}\,C and for \epsilon>0 claims that \epsilon^{-1}(C-\text{rge}\,A)\... 1answer 93 views ### Powers of compact operators Consider a Hilbert space H and a compact self-adjoint operator T : H \to H. I want to prove that all positive powers (especially fractional powers) of T are compact. From the spectral theorem, I ... 1answer 89 views ### Position operator is self adjoint Let H=L^2(\Bbb{R}) with the linear (unbounded) operator P(f)(x)=x\cdot f(x) for each x\in\mathbb{R}. Have a look at the following domain:$$D(P)=\{f\in L^2(\mathbb{R}):\int_{\mathbb{R}}{x^2\cdot|... 2answers 257 views ### Proving that$\text{ri rge}\,A=\text{ri conv rge}\,A$"If$A:\mathbb R^n\rightrightarrows\mathbb R^n$is maximal monotone,then$\text{ri rge}\,A$is convex". This is a proposition in auslender's book about the asymptotic cones. We can prove that $$\text{... 1answer 69 views ### How to use logical conjunction properly On this website in equation (20) they use$$ d \, S = a \, d \, u \land d \, v $$I have learned that \land is the truth-functional operator of logical conjunction and that such logical operators ... 1answer 47 views ### Clarification on some definitions in Operator Theory I'm trying to read this paper http://arxiv.org/abs/1206.3325 , but I'm having a lot of difficulty making sense of two phrases. The setting is L_2(\mathbb R^d). i) He mentions that for a function ... 1answer 149 views ### Proving that T(t)x is in the domain (T(t))_{t\ge 0} is a C_{0}-semigroup on a Banach space X with generator A:D(A)\subset X\to X. For k\ge 2, define$$D(A^{k}):=\{x\in D(A^{k-1})|A^{k-1}x\in D(A)\}$$I want to show that for ... 2answers 73 views ### Does there exists an operator with these properties? Consider with (\Omega,\Sigma,\mu) a \sigma-additive measure space. Is there a linear operator P \neq 0$$P : L^1(\mu) \to L^1(\mu) $$which fulfills$$ \|Pf \| \leq \|f\|, f\geq0 \... 1answer 136 views ### Extension of bounded operators between norm spaces Let$X$and$Y$be two Banach Spaces and$X_1$be a subspace of$X$. If$T$is a bounded linear operator from$X_1\to Y$, then, this is an extension of$T$from$X\to Y$such that$\|T\|_{X}\le \...
Does any continuous map between two Banach spaces is a closed map? It seems to me that it is true. Let $f:V \rightarrow W$ be a map, where $V$, $W$ are (infinite dimensional) Banach space over a a ...