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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Surjectivity of $Id-A$ for linear operator $A$ on Banach space with $\|A\|<1$

Let $X$ be Banach space and $A:X\rightarrow X$ linear opeartor such that $\|A\|<1$. It is clear that $Id-A$ is injective. Why is it also surjective?
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$A\subseteq B(X, Y)$ compact if and only if closed and $Ax$ is conditionally compact

This comes from Exercise 2 of Chapter VI in Dunford & Schwartz. I am trying to prove the following statement: A set $A\subseteq \mathscr{B}(X, Y)$ is compact in the strong operator topology if ...
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13 views

Compact Operators: Adjoint

Given Banach spaces $E$ and $F$. Consider a bounded operator: $$T:E\to F:\quad\|T\|<\infty$$ As a result due to Schauder: $$T\text{ compact}\iff T'\text{ compact}$$ How to prove this fact?
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Proof involving strong continuous semigroups

Let $T(t)$ be a $C_{0}$ semigroup on the Hilbert space $X$ with infinitesimal generator $A$ and let $\rho\in(0,1)$. I want to prove that $\displaystyle \sup_{t\ge 0}||T(t)-I||\le \rho$ is equivalent ...
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Compact Operators: Trace

Problem Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard orthonormal bases: ...
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24 views

Showing that A is NOT an infinitesimal generator

As a state space, choose $X=L^{2}(0,1)$. Let $A$ be defined as $\displaystyle Af=\frac{df}{d\zeta}$ with domain $D(A)=\{f\in L^{2}(0,1)|f$ is absolutely continuous and $\frac{df}{d\zeta}\in ...
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Finding the infinitesimal generation of a strongly continuous semigroup

Let $X$ be a Hilbert space, $A\in\mathcal{L}(X)$ and $\displaystyle T(t)=e^{At}=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}$. I have already shown that $T(t)$ defines a $C_{0}$ semigroup. But now I need ...
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How would an operator from a semigroup, act on a fixed element in Banach algebra?

Let $X$ be a Banach lattice algebra endowed with an ordering $\leq$. $T=\{T(t)\}_{t\geq 0}$ be the positive semigroup defined on $X$. $F: X_+\rightarrow X$ is a continuous mapping such that ...
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1answer
28 views

is the complexification of a finitely strictly singular operator itself FSS?

Let $X$ and $Y$ be real Banach spaces, and let $X_\mathbb{C}$ and $Y_\mathbb{C}$ denote their respective complexifications. Suppose $T:X\to Y$ is a bounded linear operator which is finitely strictly ...
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Commutant of algebra of multiplication operators

Let $L^2(X)$ be the set of Lebesgue square-integrable functions on a locally compact Hausdorff space $X$. Define $\mathfrak{A}:=\{M_f:f\in L^{\infty}(X), f=\overline{f}\}$, where $M_f$ is the the ...
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Find the spectrum of the operator $T: \ell^2(\mathbb{C}) \to \ell^2(\mathbb{C})$ defined by $(Tx)_n = \frac{x_n}{n}$

Consider the linear operator $T:\ell^2(\mathbb{C}) \to \ell^2(\mathbb{C})$ defined as $$ (Tx)_n = \frac{x_n}{n}, \quad x \in \ell^2(\mathbb{C}). $$ I can show that it is bounded with norm $\|T\|=1$, ...
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1answer
21 views

wot limit of a sequence of projections

Let $\{P_i\}$ be a net of projections on a Hilbert space , then we can show wot limit of this net is a projection, too. I saw below example of a sequence of projections which its wot limit is not a ...
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22 views

Spectrum of a bilateral shift

Let $u$ be a bilateral shift on Hilbert space $\ell^2(\Bbb Z)$. As unilateral shifts, spectrum $u$ does not contain any eigenvalue. Also $u$ is unitary, so $\sigma(u) \subset \Bbb T$ ($\Bbb T$ means ...
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1answer
34 views

Wot convergence and sot convergence

Let $\{A_n\} $ be a sequence of bounded linear operators on Hilbert space $H$ and $\langle A_n\xi,\eta \rangle \to \langle A \xi,\eta\rangle$ for $\xi,\eta\in H$ with $\|\eta\|=1$. Show that $\|A_n\xi ...
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15 views

Showing that an operator generates a unitary group

Consider the following operator on $X=L^{2}(0,1)$: $\displaystyle Af=\frac{df}{d\zeta}$ with domain: $D(A)=\{f\in L^{2}(0,1)|f$ is absolutely continuous, $\frac{df}{d\zeta}\in L^{2}(0,1)$ and ...
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42 views

The $C_0-$group generated by the operator $(Af)(x)=f'(x)+a(x)f(x)$

Consider the Banach space $L^1(\mathbb{R})$ of integrable functions $f:\mathbb{R}\to \mathbb{R}$. Consider the unbounded operator $A$ defined by $$(Af)(x)=f'(x)+a(x)f(x), \ \ \ x\in \mathbb{R}$$ for ...
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1answer
29 views

Quadratic Functional Differentiability

I would like to solve the following: Let $T$ be a self-adjoint bounded operator on a Hilbert space $H$. Consider the quadratic functional $\Phi$ defined by: \begin{equation} ...
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14 views

Binomial-like expansion for non-commutative operators

I would like to evaluate the following and express it in a compact form: $(\hat{a}^\dagger(x)+\hat{a}(x))^n\,\vert0\rangle$, where the $n^{\text{th}}$ power of the sum of the annihilation and creation ...
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1answer
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How to calculate the adjoint of an operator and its domain?

Let $A : D(A) \subset L^2(0,1) \to L^2(0, 1)$, $$D(A) = \{u \in H^2([0, 1]) : u(0) = u'(1) = 0\}$$ $$Au = u''.$$ Can someone explain how to calculate the adjoint of A, $A^*$, and the domain of $A^*$, ...
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1answer
28 views

Showing that a domain of an operator is dense in $L^2$

Let $A : D(A) \subset L^2(\Omega) \to L^2(\Omega)$, where $$D(A) = \{u \in H^2([0,1]) : u(0) = u_x (1) = 0\}.$$ Show that $D(A)$ is dense in $L^2((0, 1))$. $D(A)$ is dense in $L^2((0, 1))$ if ...
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51 views

an operator question

I know how the derivative operator $\Big(\frac{d}{dx}\Big)^n$ works. But then how does it work if I have $$\exp{\Big(a\frac{d}{dx}+b\frac{d^2}{dx^2}\Big)}f(x)$$ I thought to use $$\exp ...
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1answer
38 views

Operator matrix is invertible if and only if its determinant is invertible

Let $A,B,C,D$ are pairwise commutative operators on a Hilbert space $H$, then a necessary and sufficient condition that the operator matrix $$\begin{pmatrix} A&B\\C&D\end{pmatrix}$$ be ...
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1answer
30 views

Does this show that it is a bounded linear operator?

Let $X$ be a Hilbert space and $A\in\mathcal{L}(X)$. I want to show that $\displaystyle e^{At}:=\sum_{n=0}^{\infty}\frac{(At)^{n}}{n!}=T(t)$ defines a strongly continuous semigroup (i.e. a ...
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1answer
25 views

Differential of an operator $\phi: Mat_{2 \times 2}{\mathbb{R}} \rightarrow Mat_{2 \times 2}{\mathbb{R}}$

Let's consider an operator $ \phi: Mat_{2 \times 2}{\mathbb{R}} \rightarrow Mat_{2 \times 2}{\mathbb{R}}$ so that $A \rightarrow A^{-1}$. How to evaluate its differential? By the differential we ...
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73 views

The closed unit ball is not compact in infinite dimension spaces. Why?

We know that in finite dimension spaces the closed unit ball is compact, that is if H is a finite dimension space, then there exists an $u$ in the closed unit ball in H and $T \in \mathcal{L}(H, ...
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1answer
32 views

Calculating a norm of an operator

Let $T \in (C([a, b]))^*$, $$ T(u) = \underset{a}{\overset{(a+b)/2}\int} u(x) dx - \underset{(a+b)/2}{\overset{b}\int} u(x) dx. $$ Show that $ || T || = b - a $. We have that $$|| T || = ...
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erwin kreyszig introductory functional analysis with applications page 91 exercise 14 [closed]

Let $T$ operator from $X$ to $Y$ and $\dim X=\dim Y= n \le +\infty$ , prove that $R(T)=Y$ if and only if $T^{-1}$ exist I mean number 14
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1answer
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Application of Uniform Bounded Principle (UBP)

Let $Y$ be a Banach space, and $Z$ be a n.v.s. If $(B_n)_n\in L(E,F)$ with the property that for all $(y_n)_n\in Y$, that $\|y_n\|\rightarrow 0$, we have $\|B_n(y_n)\|\rightarrow 0$. Prove that ...
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1answer
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adjoint of an operator. on $L^2(0,1)$, $Bf(x)=\int_0^x f(t)dt$

I see that the above operator is bounded. I ended up with an argument to calculate the adjoint as follows, $$ <f,Bg>=\int_0^1\overline{f(x)} \int_0^xg(t)\,dt\,dx $$ I see $f(x)$ as the ...
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Eigenvalue dependent operator

Consider the wave equation in a Riemann metric $g^{\mu\nu}$ with spacetime off-diagonal components $g^{i0}$: ...
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17 views

Extension of a self adjoint Operator

Suppose we have a open (bounded) domain $\Omega$ in $\mathbb R^d$. And let a plane $\mathcal P$ in $\mathbb R^d$ divides the domain in two (disjoint) open sets. (say $\Omega_1$ and $\Omega_2$) Hence ...
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A sequence of strongly continuous one-parameter unitary groups

Suppose that for a sequence $\{A_n\}_n$ of bounded self-adjoint operators in a Hilbert space $\mathcal H$ we have $e^{itA_n} \to e^{itA}$ strongly, for all $t \in \mathbb R$, where $A$ is a (possibly ...
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1answer
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Essential Ideals and Denseness

Let $A$ be a $C^\ast$ algebra. I am wondering if essential ideals in $A$ are dense. It seems like they should, but I don't know how to show it.
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Discrete and Essential spectrum of Laplacian in $\mathbb R_{+}$ (with weird boundary conditions)

I am given on Hilbert Space $\mathcal H=L^2(\mathbb R_{+})$ $$ Af(x)=-f''(x) $$ and Domain of A is $$ D(A)=\{f\in H_2(\mathbb R_{+})\;\;| \;\;f'(0)+\alpha f(0)=0\} $$ for some $\alpha \in ...
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Relating Fourier transform theory on two distinct subspaces

In Fourier transform theory (on $\mathbb{R}$), three vector spaces play a very important role: $L^1(\Bbb R)$, $L^2(\Bbb R)$ and the Schwartz space $\mathcal{S}(\Bbb R)$. Arguably the nicer spaces of ...
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1answer
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+100

how to prove this epsilon-delta property for continuous functional calculus with normal elements?

Let $ A$ be a C* algebra, $f\in C([-1,1])$. Prove that for every $\epsilon >0, \exists \delta >0,$ s.t. for $\forall x \in A, x=x^*, \| x \| \leq 1$ and $\forall y \in A, \|y\| \leq 1$, we have ...
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2answers
40 views

Compact operators on Hilbert Space

I m working on the following problem: Let $K:H\rightarrow H$ be a compact operator on a Hilbert space. Show that if there exists a sequence $(u_n)_n\in H$ such that $K(u_n)$ is orthonormal, then ...
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1answer
39 views

Resolvent: Decay Behavior [on hold]

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the resolvent set: ...
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1answer
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Eigenvectors of operators on a tensor product Hilbert Space

Suppose I have finite dimensional Hilbert spaces $V$, $W$, and an operator $A$ acting on vectors in $V$ such that it has eigenvectors/values $Ax_a=\lambda_ax_a$. In the tensor product space I want to ...
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1answer
51 views

Spectral Measures: Core Lemma

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a dense domain: ...
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1answer
41 views

Spectral Measures: Scaled Spaces

Problem Given a Hillbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote its probability measures by: ...
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1answer
66 views

Convergence of the spectrum under norm resolvent convergence

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in norm resolvent sense. Since $A_n \to A$ in strong ...
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1answer
32 views

Orthonormal system of simultaneous eigenvectors

Suppose we have a commutative family of compact, self-adjoint operators on a Hilbert space. Prove that there is an orthonormal system of simultaneous eigenvectors for the family. I'm not sure how to ...
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1answer
33 views

Discrete Derivative: Closure?

Problem Given the Hilbert space $\ell^2(\mathbb{N})$. Consider the operators: $$T_0:\ell^2_0(\mathbb{N})\to\ell^2(\mathbb{N}):\quad T_0(a_k)_k:=(ka_k)_k$$ ...
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Assuming $A$ is a nonexpansion in some norm, in what norm is $A^\top$ a nonexpansion.

Consider a matrix $A \in \mathbb{R}^{n \times n}$. Consider the vector norm $\| \cdot \|_\triangle = \| F \cdot \|_1$, where $F \in \mathbb{R}^{n \times m}$ and we have $m < n$ and $F$ has ...
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1answer
19 views

Zero Tensor Product

Suppose we have a space $|\psi_1\rangle \otimes |\psi_2\rangle \otimes |\psi_3\rangle$, and operators (matrices) A ⊗ B ⊗ C acting on this Hilbert space (like in quantum mechanics). I'm trying to ...
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15 views

Riccati Finite Difference Equation:

Question: Consider the finite difference equation in Zeilberger notation: $$D_{1,x} [y] = a_0(x) + a_1 (x) y + a_2 (x)y^2 $$ Which in functional form is: $$ y(x+1) - y(x) = a_0(x) + a_1 (x) y + ...
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1answer
38 views

Spectral Measures: Analytic Elements

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the convergence radius by: ...
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20 views

Sot closure of the unit ball of a subalgebra of $B(H)$

Let $A$ be a $C^* -$ subalgebra of B(H) and $S$ be the closed unit ball of $A$. 1- $S$ is convex and bounded, so $ S = weak^* -cl ~S$. (Is it correct?) 2- By the Kapalansky density theorem, we have ...
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28 views

WOT convergence in the unit ball of B(X)

My questions is (probably) related to: On separable Hilbert space $H$, weak operator topology is metrizable on bounded parts of $B(H)$ Does the theorem quoted in the above question, together with ...