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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Wave Operators: Unitarity

This thread is Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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Dense Operators: Kernel

This thread is Q&A. Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a dense operator: ...
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If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$

If $(\lambda_n)_{n=1}^\infty$ is a bounded sequence, then there is a bounded linear operator $A$ on a Hilbert space $H$ such that $Ae_n=\lambda_n e_n$ for all $n\in \mathbb{N}$. Let ${e_n}$ be a ...
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Boundedness of $A$ in the operator equation $Au = f$ of $-\Delta u(x)=f(x)$.

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
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How to prove a set is a core for an infinitesimal generator

I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian ...
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Wave Operators: Isometry

This thread is only Q&A. Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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Isometries: Weak vs. Strong

Given a Hilbert space $\mathcal{H}$. Consider isometries: $$R_\lambda\in\mathcal{B}(\mathcal{H}):\quad R_\lambda^*R_\lambda=1$$ Then it follows: $$R_\lambda\rightharpoonup R\implies R_\lambda\to R$$ ...
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Is $B(H)$ the weak-$*$ closure of $K(H)$?

I am getting the following result: If $H$ is a Hilbert space, then the weak-$*$ closure of $K(H)$, the space of compact operators on $H$, is $B(H)$, the space of bounded operators on $H$. Is this ...
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Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$. What is a suitable $Y$ for this to hold?

Let $A$ and $B$ be bounded linear operators on a normed space $Y$ into $Y$. Show by example that $AB=I$ does not imply that $BA=I$, with $I$ being the identity operator on $Y$ Here is what I have ...
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Polar Decomposition: Adjoint

Problem Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\subseteq\mathcal{H}\to\mathcal{K}:\quad A=A^{**}$$ And its decompositions: ...
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39 views

What is the relation between the matrix of a bounded linear operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, both real or both complex, and let $\dim X = n$ and $\dim Y = m$. Let $E \colon= ( e_1, \ldots, e_n )$ be an ordered basis for $X$, and let $F ...
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1answer
30 views

Weyl sequence for closure of an operator

I'm trying to solve following exercise and need some hints. Let $A= \bar{ A_0 }$ be closure of $A_0$ - a densely defined operator. Suppose $f_n \in D(A)$ is Weyl sequence for $z \in \sigma (A)$. Show ...
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26 views

cauchy's integral formula in operator theory

Can you prove cauchy's integral formula based on the assumptions in Conway book, operator theory, VII, 4.2? 4.2. Cauchy's Integral Formula If $\mathcal X$ is a Banach space, $G$ is an open subset ...
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71 views

Closure of the Hamilton's operator $(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$ with $C_c^\infty(\mathbb{R}, \mathbb{C})$ domain

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable function bounded with its first derivative and $H$ be a Hamilton's operator such that: $$(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$$ The ...
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Uniform closure of polynomials

What is the meaning of "uniform closure of polynomials"? I have seen it in Conway's Functional Analysis book VII § 5.
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Prob. 8, Sec. 4.5 in Kreyszig's functional analysis book: The inverse of the adjoint operator is the adjoint of the inverse operator

Let $X$ and $Y$ be normed spaces, both real or both complex, let $B(X,Y)$ denote the space of all the bounded linear operators $T \colon X \to Y$, and let $T^\times$ denote the adjoint operator of ...
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How to apply Theorem 4.3-3 in the proof of Theorem 4.5-2 in Kreyszig's functional analysis book?

Here's Theorem 4.3-3 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space and let $x_0 \neq 0$ be any element of $X$. Then there exists a bounded ...
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35 views

Is the Laplacian an unbounded operator?

"The Laplacian is an unbounded operator": I read this in a book. But on Wikipedia it says: The Laplace operator $$\Delta:H^2({\mathbb R}^n)\to L^2({\mathbb R}^n) \,$$ (its domain is a Sobolev ...
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62 views

Self-adjoint operator has non-empty spectrum.

I am trying to prove, that a self-adjoint (maybe unbounded) operator has a non-empty spectrum. So far I have argued, that if $\sigma(T)$ would be empty, $T^{-1}$ would be a bounded self-adjoint ...
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36 views

tensor-operator- schatten norm

let $H_1$ and $H_2$ are hilbert spaces with dimention $n$ and $m.$ ‎assume $$\varphi:\mathcal{B}(H_1)\hat{\otimes}\mathcal{B}(H_2)\to\mathcal{B}(H_1\hat{\otimes}{H_2)},$$ which is defind as ...
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Operator not bounded below on sum of subspaces

Let $X$ be a Banach space. Say that a bounded linear operator $T\colon X\to X$ is bounded below by $\delta>0$ on $Y\subset X$ if $\|Tx\|\geqslant \delta \|x\|$ for all $x\in Y$. Is there a Banach ...
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Embedding: Extension

Problem Given Banach spaces $E_0$ and $E$ Regard dense domain: $$\overline{\mathcal{D}_0}=E_0\quad\overline{D}=E$$ Consider an embedding: ...
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Compact linear operators $S$ and $T$, show that $S(I-T)=I$ if and only if $(I-T)S=I$ and deduce that $I-(I-T)^{-1}$ is a compact operator

If $T:X\to X$ is a compact linear operator, then for any bounded linear operator $S:X\to X$ we have that $S(I-T)=I$ if and only if $(I-T)S=I$. Where $X$ is a normed space, also $T$ is bounded. With ...
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22 views

Generalization of closed range theorem

For Hilbert spaces $X$ and $Y$, the closed range theorem is ok if the operator $T:X\rightarrow Y$ has a closed range. But do we in general have \begin{equation} ...
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34 views

uniform boundedness principle - problem with understanding proof

I'm looking at the proof and there is a step I really don't get. He states that for $\parallel x-x_0\parallel<\epsilon\Rightarrow \parallel T_\alpha x\parallel$, where $T_\alpha$ Is a bounded ...
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$‎\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$ is coercive.

I am reading an article and there, author claim that $$‎L(.)=\Delta ‎^2(.)-‎\frac{‎‎\lambda‎‎}{|x|^4}(.): W^{2,2}(\Omega) \cap W^{1,2}_0(\Omega) \to W_0^{-2,2}(\Omega) ‎$$ is coercive if ‎‎$ ‎0\leq ...
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Is there a name for linear transformations $T: C(\mathbb{R}) \to \mathbb{R}^n$

Briefly, I wish to study forms that can approximate all such linear transformations $T: C(\mathbb{R}) \to \mathbb{R}^n$. Clearly the Riesz representation theorem cannot be applied in this case as ...
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Example of self-adjoint linear operator with pure point spectrum on an infinite-dimensional separable Hilbert space

Let $\mathcal{H}$ be an infinite-dimensional, complex, separable Hilbert space. Besides the well-known one-dimensional Harmonic oscillator on $\mathcal{H}=(\mathcal{L}^{2}(\mathbb{R})\,;d\mathit{l})$, ...
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Spectrum: Boundary

Problem Given a Hilbert space $\mathcal{H}$. Denote for readability: $$\Omega\subseteq\mathbb{C}:\quad\|\Omega\|:=\|\omega\|_{\omega\in\Omega}:=\sup_{\omega\in\Omega}|\omega|$$ Denote for ...
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On the square root function of matrices

Let $A, B$ be positive definite matrices and let $P$ be an orthogonal projection. If $A \leq PBP,$ does it follow that $$ A^{1/2} \leq PB^{1/2}P?$$
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Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible.

Give an example of a non-self-adjoint operator on a Hilbert space $H$ whose range is $H$ and which is not invertible. I cannot think of an example to save my life. Any solutions/hints are greatly ...
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How to extend formula for residue to functional calculus of operators

Suppose $\{X_t\}$ is a stochastic process with the covariance operator $\Gamma$ and the first $d$ eigen values are $\lambda_1\geq\lambda_2\geq \ldots \geq\lambda_d$ with eigen vectors ...
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Closedness of the range of differential operator first order

The fact that the range of gradient from $H_0^1$ to $L_2$ is closed is well known. In general we can define some kind of weak derivative in the form \begin{equation} Du=\sum_{i,j}a_{ij}\partial_i u_j ...
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Proof that every polilinear map who's domain is $R^{n_1} \times R^{n_2}… \times R^{n_k}$ and co-domain any given real normed space Y is bound.

A Polilinear map\operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
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Properties of trace-class operators

Let $X$ be a separable Hilbert space (real or complex). Let $A\in\mathcal{L}\left(X\right)$, a bounded linear operator on $X$, and suppose $B\in\mathcal{L}\left(X\right)$, which is of trace-class. ...
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1answer
74 views

How to prove this is a self-adjoint operator?

I have this operator from $H^1_0$ to $H^1_0$ defined by: $$Au(t)=\int_0^1 G(t,s) f(s,u(s))\mathsf ds$$ where $$G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to ...
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If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint.

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint. I'm lost on this problem. I don't know how to even start this. Any solutions or hints would be ...
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non compact closed range operator

Lately I've been reading Abramovich and Aliprantis' book 'An invitation to operator theory', chapter 2 (page 69) on bounded below operators. I would like to find an example of non-compact (and ...
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Wave Operators: Adjoint

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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1answer
36 views

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint. [duplicate]

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded I know that $||T||\leq 1$, but I don't know how to show this. Any solutions or ...
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Wave Operators: Cook

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}(H_\#)\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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Hamiltonian: Compactness

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the resolvent: $$R(z):=(z-H)^{-1}\in\mathcal{B}(\mathcal{H})$$ Denote compact ...
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Singular Spectrum: Criterion

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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Normal Operators: Superalgebra

Problem Given a Hilbert space $\mathcal{H}$. Consider dense operators: $$A:\mathcal{D}A\subseteq\mathcal{H}\to\mathcal{H}:\quad\overline{\mathcal{D}A}=\mathcal{H}$$ Denote the normals: ...
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1answer
54 views

Is my proof of closedness of multiplication operator corect?

I am considering an operator $A: L^2(\mathbb R , d \mu) \supset D(A)\to L^2 (\mathbb R, d\mu)$ defined by $(Af)(x)=a(x)f(x)$ for known measurable function $a$. Domain is of course all those functions ...
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prove the continuity of $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$ [duplicate]

Let $\phi\in C[0,1]$ and let $T_\phi~:C[0,1]\rightarrow\mathbb{R}$, defined as $T_\phi f=\int_0^1 f(x)\phi(x) \,dx\\$. How can i prove that it's a continuous operator?
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1answer
32 views

Prove that $Hom(V,W)\neq L(V,W)$

Let $V$ a normed space of infinite dimension and let $W\neq 0$ a normed space. Prove that $Hom(V,W)\neq L(V,W)$.
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1answer
24 views

how to define that a nonlinear operator is bounded and continuous

We always see the definition of bounded and continuous linear operator. I am wondering how to define that a nonlinear operator is bounded and continuous. Is there any book providing this definition?
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48 views

Show that $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$ is linear and continuous

Let $k:[0,1]\times[0,1]\rightarrow \mathbb{R}$ continuous and $K:C[0,1]\rightarrow C[0,1]$, given by $Kf(x,y)=\int_0^1k(x,y) f(y) \,dy\\$. Prove that $K$ is continuous. I try to see continuity in $0$, ...
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1answer
15 views

Spectral Measures: Scale Forms

This thread is only Q&A. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...