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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Showing that a function in a vector space is linear

Let $X$ be a vector space and consider a function $f : X \rightarrow \mathbb{R}$ defined for some $a \in X$ defined as $f_a (x) = a \cdot x$. (i) Prove that $f_a (x) = a \cdot x$ is a linear function....
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How do you define the inverse of an (exponential Lie) operator?

I know this is a fairly general question, but I would like to know anything I can about obtaining the inverse of an exponential of a lie operator. More specifically, I want to know how one can ...
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A necessary and sufficient condition such that product of partial isometries is a partial isometry

I'm reading the paper P. Halmos, L. Wallen, Powers of Partial Isometries, Indiana Univ. Math. J. 19 No. 8 (1970), 657–663 (http://www.iumj.indiana.edu/docs/19054/19054.asp). And I got stuck on the ...
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27 views

Generator of an analytic semigroup with a compact resolvent --> pairwise conjugate eigenvalues?

I am reviewing a paper in which the authors claim that their operator has eigenvalues $\lambda$ that are either real or pairwise conjugate (meaning that if $\lambda$ is an eigenvalue, then also the ...
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23 views

Non-strict column diagonally dominant matrix inner product

Let $A \in \mathbb{R}^{n \times n}$ be a normalized non-strict column diagonally dominant matrix, that is: $$a_{j,j} = \sum_{i \ne j} \left|a_{i,j}\right|$$ where $$0 \le a_{j,j} \le 1$$ and $$-...
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Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
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I don't see why $W^{1, 2}(\partial D)$ being compactly embedded in $L^2(\partial D)$ lets us show an operator is Fredholm of index zero.

Let $D$ be a bounded Lipschitz domain. Let $A$ be the single layer potential which maps $L^2(\partial D)$ into $W^{1, 2}(\partial D)$ boundedly. $A$ is given by: $$ A_D[\phi] = \int_{\partial D}G(x-y)...
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$Im(K)\subset Y$ closed, infinite-dimensional: $K:X\to Y$ is not a compact operator

Proposition: If $K:X\to Y$ is a bounded linear operator between two Banach spaces $X$ and $Y$ such that $\operatorname{Im}(K)\subset Y$ is an infinite-dimensional closed subspace, then $K$ is NOT ...
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Exercise involving sequence Palais-Smale and operator compact.

Let H be a Hilbert space and let K: H $\rightarrow{R}$ be $C^1$ and such that $\nabla K: H\rightarrow H$ sends bounded sets into precompact sets. Consider the functional $J: H\rightarrow R$ defined by ...
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Different ways of decomposing an exponential map

There are many decompositions of an exponential map which has two (or more) operators in the exponent (i.e. $e^{A+B}$, where $A$ and $B$ are operators). For example, the Baker-Campbell-Hausdorff (and ...
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noncommutative algebra

If we let $B$ be a noncommutative Banach algebra with unit $e$, then, obviously, $xy\not=yx$, but are they related? I suspect that the spectral radii of $xy$ and $yx$ are the same, but I couldn't ...
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Algebra $A$ and its Gelfand spectrum

Let $A$ be the set of all function $f$ on $\mathbb{R}$ of the form $$ f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R}, $$ where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...
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24 views

Commutative Banach algebra and its Gelfand spectrum

Let $A$ be the set of all functions on $\mathbb{R}^2$ of the form $$ f(t,s):=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}{a_{mn}e^{i(mt+ns)}}, $$ with the following norm: $$ \|f\|:=\sum_{m=-\...
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The completeness relation from QM in terms of inner products

I remember from QM that the completeness relation says $$ \sum_{n=1}^\infty |e_n\rangle \langle e_n | = I$$ so that $\langle x\mid y\rangle =\sum_{n=1}^\infty \langle x\mid e_n\rangle \langle e_n \...
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12 views

Commutative Banach algebra and its maximal ideal space

Let $A:=C^{(n)}([0,1])$ be the set consisting of the n-times continuously differentiable complex-valued functions. Consider $A$ with the norm $$ \|f\|:=\max\limits_{0 \leq t \leq 1} \sum_{k=0}^{n}{\...
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27 views

Why is an operator composed with its adjoint positive and stricly positive when it's invertible?

Let $V$ be a (complex) finite vector space equiped with an inner product and $T$ an operator on V. We say $T$ is positive if: $$\langle T(v), v \rangle \geq 0$$ for all $v$ in $V$. We say $T$ is ...
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Characterization of compact operators by their spectra

In any functional analysis book there is usually a section devoted to the study of the properties of the spectrum of compact operators. Is there any spectral characterization of compact (self-...
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51 views

Intuition for Fredholm operators?

Alot of the material I'm reading lately seems to mention Fredholm operators and the 'Fredholm alternative' and operators being 'Fredholm of index $0$'. Can someone give me a high level overview of ...
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45 views

Operator theory to study a difference equation

I'm not an expert in operator theory (so I'm going to be very informal sorry), but I would like to be given some advice about a problem I have. Let $f$ be a function defined in $C^{\infty}(\mathbb{R})$...
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continuous and sequentially continuous

If an operator $T: A\rightarrow B$ satisfying for every sequence $\{X_n\}$ weakly converging to $X$, we have $TX_n \rightarrow TX$ in weak topology. Then, is $T$ weak-weak continuous? And in the WOT/...
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R.Douglas “Banach Algebra Technique Operator Theory” - Chapter 2 issue

Just before 2.37 Corollary (Spectral Mapping Theorem) Douglas says: If $\varphi (z)= \sum_{n=0}^\infty a_nz^n$ is an entire function with complex coefficients and $f$ is an element of the Banach ...
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For $p(x)\in \Bbb{C}[x]$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $0\le k\le n-1$, show that $p(\lambda)=0\Rightarrow \lambda\in [0,1]$

For a complex polynomial $p(x)\in \Bbb{C}[x]$ of degree $n$ such that $\int_{0}^{1}p(x)x^kdx=0$ for $1\le k\le n-1$, show that $p(\lambda)=0$ means $\lambda\in [0,1]$. I haven't come by any ...
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About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $ P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
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Prove or disprove: $\{t^{2k}\}_{k=0}^{\infty}$ complete in $L_2[-1,3]$

Is $\{t^{2k}\}_{k=0}^{\infty}$ not complete in $L_2[-1,3]$?(Here, completeness of a system is equivalent to the density of its span) Obviously many polynomials in the domain will be irreleant, but I ...
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3answers
81 views

$\ker ST=\ker T$

Let $S$ and $T$ be linear maps between vector spaces such that the composition $ST$ makes sense. Clearly, $\ker ST\supseteq \ker T$. The two instances that come to my mind for having an equality in ...
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37 views

What is the Hilbert adjoint operator of this bounded linear operator?

Let $H$ be a Hilbert space, and let $z \in H$. Let $T_z \colon H \to K$, where $K$ is the field of scalars for $H$ and $K$ is either $\mathbb{R}$ or $\mathbb{C}$, be defined by $$ T_z (x) \colon= \...
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Finding closure of image of operator

I'm working on an old exam problem: Define for $u \in C^2([-1,1])$ the operator $L$ by $[Lu](x) = - \frac{d}{dx} \left( (1-x^2) u'(x) \right)$. Set $\Omega = \{ Lu \mid u \in C^2([-1,1]) \}$. Find the ...
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Closure of an operator

I am wondering what is the closure of the domain of the operator $A_0:D(A_0)(\subset H)\to H$in $H=L^2(0,1)$ $$A_0= f^{(4)}-f^{(6)}$$ $$D(A_0)=\big\{ f\in H^6(0,1)\cap H_0^3(0,1) |f^{(3)}(1)=f^{(4)}(...
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25 views

If $U$ is a vector subspace of a Hilbert space $H$, then each $x∈H$ acts on $U$ as a bounded linear function $〈x〉$. Is $x↦〈x〉$ injective?

If $H$ is a $\mathbb R$-Hilbert space, then the duality pairing $$\langle\;\cdot\;,\;\cdot\;\rangle_{H,\:H'}:H\times H'\;,\;\;\;(x,\Phi)\mapsto\Phi(x)$$ can be considered as being a mapping $H\times H\...
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Relationship between the distributional Laplacian and the weak Laplacian

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\langle\;\cdot\;,\;\cdot\;\rangle$ denote the $L^2(\Omega)$- or $L^2(\Omega,\mathbb R^d)$-inner product (depending on the context) $\mathcal ...
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What is the correct definition for positive operator and positive definite operator?

As far as I know those operators are defined as follows: Positive operator is an operator $L: H\rightarrow H$ such that $\langle L\textbf u|\textbf u\rangle \geq0$ for all $\textbf u \in H$ and the ...
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$T^p$ increases to $T$ in strongly operator topology or not.

In a Hilbert space $H$, let $T$ be a positive operator on $H$ with $\|T\|_\infty\le 1$. Then, obviously, $T^p$ is increasing as $p$ decreases to 1. But I am not sure whether $T^p$ increases to $T$ in ...
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Has the distributional Laplacian $\Delta f:C_c^\infty(\Omega)'\to C_c^\infty(\Omega)'$ a unique extension in $H_0^1(\Omega)'$?

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D:=C_c^\infty(\Omega)$ and $$H=\overline{\mathcal D}^{\langle\;\cdot\;,\;\cdot\;\rangle_H}\tag 1$$ with $$\langle\phi,\psi\rangle_H:...
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Find the spectrum of an operator related to Fourier series

As an exercise, I was told to find the spectrum of the bounded operator $K\in B(L^2[-\pi,\pi])$ defined by $$K\varphi (t)=t\int_{-\pi}^\pi\varphi (x)\cos (x)dx+\cos t\int_{-\pi}^\pi x\varphi(x)dx.$$ ...
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When is orthogonal projection compact? [duplicate]

Let $M$ be a closed subspace of a Hilbert space $H$. Let $P$ be the orthogonal projection on $M$. I was told to find the eigenvalues and eigenvectors of $P$ and moreover say when it is compact. Since ...
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Find eigenvalues and eigenvectors of infinite symmetric matrix of powers of two

Let $a_n=2^{-n}$. What are the eigenvalues and eigenvectors of the $\ell^2$ operator represented by the infinite matrix below? $$A=\begin{pmatrix} a_1 & a_2 & a_3 & \dots \\ a_2 & a_3 &...
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Simple norm inequality

Trying to follow the comments to this question I am struggling very much to understand how to simplify $\|Ax\|_2=\sup_{\|x\|_2=1}\sqrt{\sum_i(\sum_ja_{ij}x_j)^2}$ to arrive at an $x$-free bound. Can ...
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Prove multiplication by sequence is a compact operator

Let $c_0(\mathbb N)$ be the space of sequence in $\mathbb C$ whose limit is zero, equipped with the $\ell^\infty$ norm. Let $u_n$ be a sequence in $\mathbb C$ and define the operator $A$ taking a ...
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$ (k\otimes h^\ast)^\ast=h\otimes k^\ast$?

Let $H,K$ be Hilbert spaces with $h\in H,k\in K$. Let $k\otimes h^\ast(g)= \left\langle g,h \right\rangle k$. I'm supposed to prove $ (k\otimes h^\ast)^\ast=h\otimes k^\ast$, but I don't see how this ...
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Proving an operator is compact exercise

Suppose $(a_{ij})_{i,j\in \mathbb N}$ satisfy $\sum_{i,j}|a_{ij}|^2<\infty$ and define $A:\ell ^2 \rightarrow \ell ^2$ by $(Ax)_i)=\sum _j a_{ij}x_j$. I need to prove $A$ is compact. Unfortunately,...
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1answer
69 views

Determining whether equality $ \|T v\| = \|T\| \cdot \|v\|$ is possible

As an exercise, I'm supposed to determine whether for the operators $A_\lambda:g(t)\mapsto \sqrt\lambda g(\lambda t)$ on $C[0,1],\lambda\in (0,1)$ and $T_f:g\mapsto g(t)f(t)$ on $L^2$ it is possible ...
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Trace norm of a triangular matrix with only ones above the diagonal

For $n\in\mathbb N^*$, we consider the triangular matrix $$ T_n = \begin{pmatrix} 1 & \cdots & 1 \\ & \ddots & \vdots \\ 0 & & 1 \end{pmatrix} \in M_{n,n}(\mathbb R) \,. $$ ...
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For which $\alpha$ is this integral operator compact?

I have $Q\subset\mathbb{R}^n$ $Af(x)=\int_QK(x,y)f(y)dy$ , with $K(x,y)=\frac{K_0(x,y)}{|x-y|^\alpha}$ and $K_0\in C(Q)$ I want to estimate using an operator $A_Mf(x)=\int_QK_M(x,y)f(y)dy$ where, $...
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32 views

Exercise on the operator $g(t)\mapsto \sqrt \lambda g(\lambda t)$

I have the linear operator $A_\lambda :g(t)\mapsto \sqrt \lambda g(\lambda t)$ on $C[0,1]$ with $\lambda\in (0,1)$ (which extends to $L^2[0,1]$). I need to find its adjoint and then prove that while $...
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Fibers of unbounded linear functional are dense

I'm supposed to prove that if $f$ is a discontinuous linear functional $H\rightarrow \mathbb C$, each of its fibers $f^{-1} \left\{ \alpha \right\} $ is dense. I already know the kernel, i.e $f^{-1} \...
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1answer
40 views

Showing space is complete and functional is bounded

I've got two questions. (1) Define $ \|f\| := |f(0)| + \max \limits_{0 \le t \le 1} |f'(t)|$ to be a norm on $C^1([0,1])$. (I have shown it's indeed a norm). Show that $(C^1([0,1]), \|\cdot\|)$ is ...
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45 views

Universal $C^*$ algebras

It is known that the $C^*$-algebra $\mathcal U$ generated by bilateral shift $\ell^2 (\mathbb Z) \ni e_k \mapsto e_{k+1} \in \ell ^2(\mathbb Z)$, is a universal $C^*$ algebra generated by unitary: for ...
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40 views

Spectrum of the Resolvent of a Self-Adjoint Operator

Let $\mathcal{H}$ be a Hilbert space, and $A$ a self-adjoint operator with domain $D_{A} \subseteq \mathcal{H}$. Assume that $\lambda_0 \in \rho(A)$, where $\rho(A)$ is the resolvent set of $A$. For ...
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49 views

Definition of the Laplacian as an operator from $H_0^1(\Omega)$ to $H_0^1(\Omega)'$

Let $d\in\mathbb N$ $\Omega\subseteq\mathbb R^d$ be open $\mathcal D(\Omega):=C_c^\infty(\Omega)$ $f\in L^2(\Omega)$ and $$\langle f\rangle:=\left.\langle\;\cdot\;,f\rangle_{L^2(\Omega)}\right|_{\...
2
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1answer
41 views

Norm of the Resolvent

Let $\mathbb{H}$ be a Hilbert space, $A$ a self-adjoint operator with domain $D_{A}$, $R_{A}$ the resolvent of $A$, and $z$ a point in the resolvent set $\rho(A)$. How could you prove the inequality \...