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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PDE and Linear operators

Consider a linear PDE which can be seen as L[u] = 0, where L is a linear differential operator on u. Is there any theory that tries to study the PDE by sutdying the kernel of L?
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Why do mathematicians say that “let an operator be represented by a matrix” instead of operator is the matrix?

For example, look at this sentence from Perko's text on dynamical system "It follows from Cauchy Schwarz inequality that if $T \in L(R^n)$ is represented by the matrix $A$ with respect to the ...
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Proposed proof Decomposition theorem

Let $\mathcal{A}$ be a unital Banach algebra. I want to prove that if $a \in \mathcal{A}$ and spectrum $\sigma(a) = \sigma_{1} \cup \sigma_{2}$ where $\sigma_{1} \cap \sigma_{2} = \emptyset$, ...
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Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping.

Let $Y$ be a finite dimensional normed space, $X$ a normed space, and $T: X \to Y$ a surjective linear operator. Show that $T$ is an open mapping. I think if I can show that $T(B_X)$ contains an ...
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If a subspace of $X^*$ is weak*-dense, does it separate points?

Here $X$ is some normed space. I know the converse is true, but I don't know a proof for the other direction. That is, if $F\subset X^*$ is a subspace that is weak*-dense how would one show that ...
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Norm of derivative of rank one projector

Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation \begin{equation} i\partial_t\phi(t)=-\Delta\phi(t)+(V*|\phi|^2)\phi(t) \end{equation} inside the Hilbert space $L^2(\mathbb{R}^d)$. ...
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Understanding how Nehari's problem connects with robust stabiliziation and Nevanlinna-Pick

I'm reading Young's "An Introduction to Hilbert space". In chapter 15 he writes about robust stabilization in control theory and ends with that this boils down to an interpolation problem called the ...
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Infinite dimension left shift operator over the complex vector space

Let $S$ be the left shift operator over the infinite complex vector field. Show that $$ \text{null}(S-I)^3=\text{span}\{(1,1,1,1,\ldots),(0,1,2,3,4,5,6,\ldots),(0,1,4,9,16,25,.....)\}. $$ To start I ...
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exercise, Hahn-banach theorem

I have this exerciose: Let $\Omega$ be a normed space. Prove that $\Omega$ is seperable if $\Omega^*$ is. It is in the chapter with the Hahn-Banach theorem, so I think I should use that ...
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Prove that $A\geq I$ implies that $A$ is invertible.

Here's the question: Let $A$ be a positive operator on a (possibly infinite dimensional) Hilbert space. Let $I$ denote the identity operator. Suppose that $A \geq I$, which is to say that $A - ...
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Norm of a Positive Operator

For a positive operator $A\in B(\mathcal{H})$ on a complex Hilbert space $\mathcal{H}$, I want to prove that $\|A\|=\sup\{\lambda: \lambda \in \sigma(A)\}$, where $\sigma(A)$ is the spectrum of $A$. ...
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$\mathcal{A}+K$ is norm-closed where $\mathcal{A}$ is a $C^*$-algebra and $K$ is the compact operators.

Let $\mathcal{A}\subset B(H)$ be a unital $C^*$-algebra and let $K$ be the closed ideal of compact operators. I need to show that $\mathcal{A}+K$ is also a $C^*$-subalgebra of $B(H)$. I am stuck at ...
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Showing that $\overline{ T(B_{X,1}(0))}$ in a Banach space $Y$ is convex.

Consider the following problem. If I have Banach spaces $X,Y$, and an operator $T\in B(X,Y)$, how would I show that, "If $T\in B(X,Y)$, and $p\in\overline{ T(B_{X,1}(0))}$ then $-p\in\overline{ ...
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Maximal ideal space and quotient space in abelian Banach algebra

I have a short question regarding operator algebras. Given an abelian Banach algebra $\mathcal{A}$. Assume that $\phi \in \big\{ \phi : \phi \text{ is a non-zero linear multiplicative functional} ...
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How to show that the heat trace converges

Suppose $M$ is a compact Riemannian manifold of dimension $d$ without boundary, with Laplace Beltrami operator $\triangle$. We know that the spectrum of $\triangle$ (defined appropriately as a self - ...
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2answers
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Point on the proof that the inverse operator of $I-T$ is given by $(I-T)^{-1}=\sum_{k=0}^\infty T^k$

Let $X$ be a Banach space and let $T\in B(X)$ be such that $\|T\|\lt1$. Suppose then we have the operator $I-T$ and we want to show that its inverse operator $(I-T)^{-1}$ is given by the following ...
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Left shift operator, linear algebra

Define $S$ to be the left shift operator from $\mathbb{C}^\infty$ to $\mathbb{C}^\infty$. Describe the kernel of $S^2-S-I$, what is ts dimension? Give a Jordan basis of S for the restriction to this ...
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19 views

C*-algebra generated by a non invertible normal element

Let $A$ be a C*-algebra and $x\in A$ be a non-invertible normal element. By functional calculus, we know $$C^*(x,1)\simeq C(\sigma(x))$$ Where $\sigma(x)$ means the spectrum of $x$. I need to ...
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Showing that a certain operator maps to $\mathscr{C}([0,1])$

I'm considering the operator $T$ given by (Tf)(x)=$\int_0^1k(x,y)f(y)dy$ with $dom(T)=\mathrm{L}^1([0,1])$, where $k\in\mathscr{C}([0,1]^2)$ and want to proof that it maps to $\mathscr{C}([0,1])$. I ...
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How to find a unitary operator U such that U L U=L*??

I am reading the "integral equations--- a practical treatment, from spectral theory to applications" by David Porter and David S. G. Stirling. In page 293 there is an operator $K_\alpha$ defined by: ...
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Boundedness of an operator in $L^p$ space

Let's define the following operator $$\mathcal{J}_\epsilon:=(I-\epsilon\Delta)^{-1},$$ where $\epsilon>0$ and $\Delta$ is the Laplacian. We know that $$\Vert\mathcal{J}_\epsilon ...
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Domain of double adjoint

For $T$ a densely defined linear, not necessarily bounded operator on a Hilbert space $\mathscr{H}$, and $T^{**}$ the adjoint of $T$'s adjoint, I read somewhere that $Ran(T)=Ran(T^{**})$. Is that ...
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Show that a linear operator is a projection.

Say I have the following linear operator, $$S(g)(y)=g(y)-\int_0^1g(x)dx$$ on the space $C([0,1])$ where $g$ is a function therein, and that I want to show that $S$ is a projection, that is to say ...
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Show that the operator $S\in B(Y)$ (the set of bounded linear operators from $Y\to Y$)

I am having trouble with the following problem: "Suppose we have an operator $S:Y\to Y$, where, $$S(g)(y)=g(y)-\int_0^1g(x)dx$$ and where we have $Y=C([0,1])$, equipped with the uniform norm ...
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Proving Toeplitz matrix defines bounded operator on $ l^2 $

I should first mention this: I have asked this question previously but I only got a partial answer that does not suit the actual assumptions but only the related ones, it reads as follows: Define ...
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1answer
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Is this operator positive?

When is the operator $\begin{pmatrix} S^*S & \sqrt{2} S \\ \sqrt{2}S^* & S^*S \end{pmatrix}$ positive, where $S \in B(H)$ is of norm $\sqrt{2}$.
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Looking for a general approach to proving compactness of linear operators.

Preparing for my exam in functional analysis, I often have to prove that certain explicitly given operators are compact. I now have a decent amount of operators of which I can prove the compactness, ...
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need example of a strictly positive compact operator on a Hilbert Space

Give an example (if it exists) of a strictly positive compact operator on an infinite dimensional Hilbert Space
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Some claims about spectra

I see the following equalities used sometimes, but couldn't find proofs. How is it done? $$\overline{Ran(L-\lambda\mathbb{1})}^{\perp}=ker(L^*-\overline{\lambda}\mathbb{1})$$ and ...
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Infinite matrix SIGNIFICANT PROGRESS BUT STUCK defines compact operator on $ l^2 $

I have this question with a given clue I cannot decipher in Functional Analysis stating: Define the infinite matrix $ A = [a_{ij}]_{i,j=1}^{\infty} $ using a sequence $ \{ \alpha_n ...
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If $||A^N|| < 1$ then is ||A||<1?

following a previous question in functional analysis I asked Let $X$ be a banach space and $A$ is a bounded linear operator on $X$, and there exists some natural $N$ for which $ ||A^N|| < 1 $. ...
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If $||A^N|| < 1$, is $I-A$ invertible?

i have been given this question in functional analysis saying: Let $X$ be a banach space and $A$ is a bounded linear operator on $X$, and there exists some natural $N$ for which $ ||A^N|| < 1 ...
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Square root of Self-Adjoint Operator

I have $H=L^2(0,2)$ and $Aw=w^{(4)}, D(A)=H^4(0,2)\cap H^2_0(0,2)$, this operator is non-negative and self-adjoint because $A$ is monotone maximal, $R(I+A)=H$. A form $t(u,v)=(Au,v)_{H}$ is closable, ...
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How to define Biharmonic operator for second order Sobolev spaces and show that it is continuous?

I am studying an article where the authors assume that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Somewhere in the paper we have $$ \Delta^2 (\cdot) : W^{2,2}(\Omega) \to W^{-2,2}(\Omega) $$ (This ...
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Root Square Convergence

I'm trying to solve this problem: Prove that if $A_n\geq 0$, $A_n\rightarrow A$ in norm, then $\sqrt{A_n}\rightarrow\sqrt{A}$ in norm. Where everything are bounded and linear operators in a Hilbert ...
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product of measure induces ? of linear operators

Given two measures $\mu_1$ and $\mu_2$, we can form their product. For simplicity, let's assume we're working over $\mathbb{R}$, and that these measures are absolutely continuous wrt Lebesgue, so ...
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In which sense an expansive matrix is expansive

Let $A\in M_n(\mathbb{R})$. $A$ is called an expansive matrix if every eigenvalue of $A$ is strictly bigger than 1. From the name "expansive", one may deduce that (the space of square expansive ...
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Inequality with operators

Hi I have A and B two bounded operators, I used the inequality $$\||A|-|B|\|\leq\|A-B\|$$ where, $|A|=(A^*A)^{\frac{1}{2}}$ and $A^*$ the adjoint. But my professor said this is not necessarily true, ...
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Sobolev spaces and symmetric operators

I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following: Suppose we are ...
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Matrix whose eigenvectors are Hermite polynomials

I first constructed a symmetric matrix as the Laplacian operator, and its eigenvectors are a series of harmonics functions as expected. I programmed it and convinced myself. The matrix looks like: $$ ...
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$(\partial_{tt}+\partial_t-\nabla^2)f(r,t)=0$

Hi I am trying to find the kernel of the linear differential operator $D$ $$ D\equiv\partial_{tt}+b\partial_t-a\nabla^2,\quad a,b>0. $$ We have $$ \nabla^2\equiv ...
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How to show that the set of square matrices $R^{n \times n}$ is complete under the operator norm $\|A\| = \sup\limits_{\|x\|\leq 1} \|Ax\|$

I want to show that the set of square matrices $R^{n \times n}$ is a Banach algebra with property $\|AB\| \leq \|A\|\|B\|$. I have already showed that $R^{n \times n}$ is a linear space and it is a ...
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The Multiplier Algebra of the Hardy Hilbert Space

Let $H$ be a Hilbert space of analytic functions. Define the multiplier algebra of H in the following manner: $M(H)=\{ \phi \in H : \phi h \in H, \forall h \in H \}$ It is mentioned in countless ...
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Finding Linear Operator for a given Basis

Consider a linear operator $$L: \lbrace f: \Bbb{R}\rightarrow \Bbb{R} \rbrace \rightarrow \lbrace f: \Bbb{R}\rightarrow \Bbb{R} \rbrace $$ For example $$ L(f) = f(x+1) - f(x)$$ Define the ...
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How to apply Picard-Lindelof existence and uniqueness theorem for autonomous LTI dynamical system $\dot x = Ax$?

In nonlinear dynamical system, we have the picard-lindelof existence and uniqueness principle which guarantees existence of unique solution to problem of the type $\dot x = f(x,t), x(0) = x_o$ ...
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An equivalence related to determining the point spectrum of multiplication operator

Given a semifinite measure space $(X,M,\mu)$, a number $1\le p\le\infty$ and a bounded measurable function $m:X\rightarrow\mathbb{C}$, how do I formally prove that $$\forall\varphi\in ...
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minimal projections in matrix-algebras

Consider $A=\{ \begin{pmatrix} T & 0 \\ 0 & T \end{pmatrix}: T\in M_2(\mathbb{C})\}\subseteq M_4(\mathbb{C})$ and $p= \begin{pmatrix} 1 & 0&0&0 \\ 0 & 0&0&0\\0 & ...
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Polar decomposition theorem for symplectic and orthogonal Banach Lie groups in infinite dimensional settings

Could you please help me to understand the polar decomposition theorem for $Sp(H, J_Q)$ and $O(H,J_R)$ where $H$ is infinite dimensional separable Hilbert space and $J_R$ and $J_Q$ stands for ...
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Weakly convergent subsequence under continuous operator

Suppose we have two Hilbert spaces $H_1,~ H_2$, a linear continuous operator $T:H_1 \to H_2$ and a weakly convergent sequence $u_k\rightharpoonup u$ in $H_1$. Is $Tu_k \rightharpoonup Tu$ in $H_2$ ...
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1answer
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Bounding a linear map from $L^q$ to $C_c$

Let $C_0((0, \infty))$ be the set of all functions such that $\lim_{x \to \infty} f(x) = 0$ and $\lim_{x \to 0} f(x) = 0$, this is a Banach space under the $\sup$ norm. Now if we fix a $p$, $1 < ...