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 does the set of an hermitian operator's eigenfunctions spans the functions space

During a discussion about linear hermitian operators, my professor claimed that if a linear operator $M$ is hermitian under a certian set of conditions, then genrally any function that fulfills these ...
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Commutativity and norms of specific operators (Problem 2.7.10 in Kreyszig's functional analysis book)

This is Problem 2.7.10 from Erwin Kreyszig's Introductory Functional Analysis with Applications. Let $C[0,1]$ denote the normed space of all (real- or complex-valued) functions defined and ...
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projections on a Hilbert space [on hold]

Let $p, q$ be projections on a Hilbert space $H$. Then the following conditions are equivalent: $p < q.$ $pq = p.$ $qp = p.$ $p(H) \subseteq q(H).$ $\|p( x )\| < \| q ( x ) \|$, $(x \in H)$ $q ...
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Density and Fredholmness

Let $X$ be a Banach Space and $Y$ a dense subset of $X$. An operator $T:X \to X$ is said to be Fredholm if it has closed range, $\dim \ker(T)<\infty$ and $\dim coker(T) < \infty$. Here is my ...
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Problem 8, Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 8 in the Problem Set following Section 2.7 in Erwine Kryszeg's Introductory Functional Analysis With Applications: Show that the inverse $T^{-1} \colon R(T) \to X$ of a bounded ...
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Problem 2.7-9 in Erwine Kryszeg's Introductory Functional Analysis With Applications

Here is Problem 9 in the Problem Set following Section 2.7 in the book Introductory Functional Analysis With Applications by Erwine Kryszeg: Let $C[0,1]$ denote the set of all (real- or ...
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If $H$ is a one-dimensional Hilbert space then the zero representation of a C*-algebra on $H$ is irreducible.

It says on page 143 of Murphy's $C^*$-algebras and operator theory that if $H$ is a one-dimensional Hilbert space then the zero representation of any C*-algebra on H is irreducible. What is the zero ...
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$s \in L^{1}(H)$ $\iff$ $s=\sum_{i=0}^\infty x_{i} \otimes y_{i} $ [on hold]

Let $H$ be a separable Hilbert space, and let $L^1(H)$ be the space of trace-class operators on $H$. I'd like to prove that $s\in L^{1}(H)$ if and only if there exists $\{ x_{i} \} , \{ y_{i} \} ...
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The resolvent of a differentation operator on $C[a,b]$

Consider a densely defined operator $A : C[a,b ]\rightarrow C[a,b ]$, $$Au=u^{\prime}$$ with domain $$D(A)= \{ u\in C^1[a,b]: u(b)=ku(a) \}$$ for some $k>0$. I have to find $R_A(\lambda)$ for ...
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The ideal generated by a non-compact operator

I wanted to find a quick proof of the following well-known fact. Since I couldn't easily find a reference, I provide a proof below. Let $H$ be a separable Hilbert space, and $J\subset B(H)$ be a ...
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Resolvent operator

Let's consider the following operator on $L^2(\mathbb{R}^3)$ $$A(t)=\Delta+b(t,x)\cdot\nabla$$ where $\Delta$ is the Laplace operator and $b(\cdot,\cdot)$ a smooth vector field. How to compute the ...
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1answer
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If $I$ is a closed ideal in a C*-algebra $A$ and $J$ is a closed ideal in $I$ then $J$ is an ideal of $A$

The following is a remark of Murphy's C*-algebras and operator theory: . I do not know why he uses approximate unit. I think for $a\in A$ and $b\in J^+$, we have $b\in I$ and $b^{1/2}\in I$($I$ is ...
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Problem 2.7.6 in Kreyszig's Introductory Functional Analysis with Applications

Suppose that $X$ and $Y$ are two normed spaces over the same field ($\mathbb{R}$ or $\mathbb{C}$). Show that the range of a bounded linear operator $T \colon X \to Y$ need not be closed in $Y$. ...
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Can a Local Fractional Differential Operator exist?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$. The derivative of $f$ is defined pointwise, and we say that $f$ is differentiable if the derivative exists in each point. Higher order derivatives are ...
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Dynamics: Schwinger-Dyson-Expansion

Given a C*-algebra $\mathcal{A}$ Consider a free generator $\delta_0:\mathcal{D}_0\to\mathcal{A}$ with $\overline{\mathcal{D}_0}=\mathcal{A}$. Introduce a perturbation ...
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Resolvent set of derivative

i don't understand how to find spectrum and resolvent set $A : C[ab ]\rightarrow C[ab ]$ $D(A)= ( u\in C^1[ab]: u(b)=ku(a) )$ for some $k>0$ need to find $R_A(\lambda)$ and $\sigma (A)$. I can ...
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Extending $*$-isomorphisms between $*$-algebras to cross products.

Let $G$ be a discrete countable group and suppose I have two $G$-$C^*$-algebras $A$ and $B$ such that there exists a $G$-equivariant isometric $*$-isomorphism $\varphi \colon A \to B$. One can extend ...
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1answer
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$\phi(A^+) \subset B^+$ when $\phi: A\to B$ is an isometric linear map

Let $\phi: A\to B$ be an isometric linear map between unital C*-algebras $A$ and $B$ such that $\phi(a^*)=\phi(a)^* (a\in A)$ and $\phi(1)=1$. Show that $\phi(A^+) \subset B^+$. Clearly $A^+ = \{a^*a ...
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Norm of an operator and eigenvalues

I have $K\colon L^2(0,T) \to L^2(0,T)$ a Hilbert-Schmidt integral operator (and so $K$ is linear, bounded, compact and self-adjoint) and I have obtained its eigenvalues and eigenvectors. From them, I ...
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1answer
17 views

Show that hermitian element $h=\sum p_n/3^n$ generates $ C_0(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space, and suppose that the C*-algebra $C_0(\Omega)$ is generated by a sequence of projections $(p_n)_{n=1}^{\infty}$. Show that the hermitian element ...
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Spectra of periodic Schrödinger equations

This question might be a little bit physics-related, but I kind of have a deep interest to ask this here, cause I would like to get an idea of the Mathematics behind the things I just covered in my ...
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1answer
36 views

Self-adjointness

In another thread it was claimed that the operator $O : \operatorname{dom}(O) \subset L^2(-1,1) \rightarrow L^2(-1,1)$ is self-adjoint. $$Of(x)= \frac{f(x)}{{1-x^2}}$$ It is obvious that $$\langle O ...
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Preserve self-adjoint properties

I was thinking about this problem recently: Let $T$ be a self-adjoint operator on $L^2((-1,1),d x)$. Now you define an operator $G$ by $G(f) := T(\frac{f}{(1-x^2)})$ with $\operatorname{dom}(G):=\{f ...
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32 views

How to show that the operator $T(\{x_n\})=\{n x_n\}$ has closed graph?

Consider the subspace $$D=\left\{x\in \ell^2 \ \big|\ \sum_{n\in\mathbb N} n^2 |x_n|^2<\infty\right\}$$ of $\ell^2$, and let $T:D\to\ell^2$ be defined by $T(\{x_n\})=\{n x_n\}$. I need ...
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Definition of well-defined for special case

I have a question about what well-defined means in a certain case. For an operator from $X$ to its dual $X^{*}$, say $A:X \rightarrow X^{*}$,why does the definition of $A$ being "well−defined" seem ...
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31 views

Vector question involving an operator!

So, here's the problem: An operator H capable of operating on vector x, is defined in terms of a given vector a by: H x=(a * x) where $*$ representes vector product Given that ...
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Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
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1answer
21 views

Holomorphic Functional Calculus vs Borel Functional Calculus

I am currently learning about different kinds of functional calculus and I was wondering if I could get something cleared up. The first type of functional calculus we learned about was holomorphic ...
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$C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection

I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...
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Strong derivative of a compound map

I find the strong Fréchet derivative of $\Phi(h,\psi(h))$, where $\Phi:T_0\times T_\xi\to Y$ with $T_0, T_\xi, Y$ Banach spaces and $\psi:T_0\to T_\xi$ is strongly differentiable in $0$, evaluated in ...
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Closed restriction of an unbounded self-adjoint operator

Suppose $(A\,;\mathcal{D}(A))$ is an unbounded self-adjoint linear operator (obviously, $\mathcal{D}(A)$ must be dense) on a Hilbert space $\mathcal{H}$. Suppose $\mathcal{D}(C)$ is a proper dense ...
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Iterating average

If $f$ is a continuous function $[0,1]\to \mathbb R$, we define a linear application $T$ as follows $$T(f)(x)=\begin{cases} f(0) & \mathrm{if }~ x=0 \\[0.2cm] \displaystyle ...
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Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a ...
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Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
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Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
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Square root of unbounded operator

Let $T: \operatorname{dom}(T) \subset H \rightarrow H$ be a positive self-adjoint unbounded operator, then I want to define a UNIQUE(!) operator $A$ such that $A^{*}A = T$. Actually, this construction ...
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Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...
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Finite rank approximation of bounded operators on Hilbert space

Let H be a (finite dimensional) Hilbert space. The approximation property states that every bounded operator from H to itself can be approximated by a sequence of finite rank operators. My question ...
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Spectrum of adjoint operator

Let $X$ be a hilbert space and $T\in L(X)$ Show that: (i) $\sigma_c(T^*)=(\sigma(T))^*$ (ii) $\sigma_r(T)=((\sigma_p(T^*))^*)$\ $\sigma_p(T)$ (i): $"\subset"$ Let $\lambda\in\sigma_c(T^*)$ then ...
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operator ranges of a closed linear operator

I have a closed linear operator T on a Hilbert space H. Also I have that range R(T^n) is closed for some n $\in$ N and for non-zero $\lambda$, R(T-$\lambda$I) $\cap$ R(T^n) is closed and ...
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CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...
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residual spectra of adjoint operator.

I saw an interesting claim saying "If $λ$ were in the residual spectrum of $T$, then $\overline{λ}$ would be in the point spectrum of $T^*$. Is there a short proof for this claim? recall point ...
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1answer
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Position Operator on $l^2(\mathbb{Z})$

I'm very familiar with the position operator $(Q\varphi)(x)=x\varphi(x)$ on $L^2(\mathbb{R^d})$, but I'm trying to figure out how to interpret the same operator on $l^2(\mathbb{Z})$ (the space of ...
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Multilateral shift operators

Apparently, the operators $u(e_n) = e_{n+1}$ and $u^\ast (e_n ) = e_{n-1}$ are called "unilateral shift oeprator". Since they have to be called that (instead of calling them "the shift operator") ...
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Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
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1answer
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Approximation in $V^{**}$

We know that for a vector space $V$ there is a natural map $\iota\colon V\to V^{**}$ sending $v$ to $v^{**}$. When $V$ is a normed space, $\iota$ is an isometry, however it may not be surjective. My ...
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tensor products of Banach space

Let $E_{1},\cdots, E_{n}$ be Banach spaces; $n\in\mathbb{N}$ and $\mathbb{R}$ be a real numbers and $E\widehat{\otimes}\mathbb{R}$ be a completion tensor product. We have the fact that ...
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Analytic vectors of self-adjoint unbounded operators

I am working with self-adjoint unbounded linear operators on infinite-dimensional separable Hilbert spaces, and I would like to know some references regarding analytic vectors, especially with respect ...
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29 views

Does every continuous operator between normed spaces map bounded sets to bounded sets?

Suppose you have a continuous operator $A$ between two normed spaces $X$ and $Y$. Does it follow that this operator is bounded in the sense that it takes bounded sets to bounded sets, given that $A$ ...
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Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...