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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Spectral raius for linear compact maps

Prove or disprove the following assertions for a linear map $C$ from a Banach space $X$ into itself: a) If C is compact then its spectral radius equals the maximum of the absolute value of $C$ Im ...
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448 views

Operators on $C([0,1])$ that is compact or not.

For $f\in C([0,1])$ set $$Hf(x) = \frac{1}{x}\int_0^x f(t)dt.$$ a) Show that $H$ is a bounded operator from $C([0,1])$ into itself which is not compact. b) From a) it follows that $H$ induces a ...
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195 views

Compute spectral/projection-valued measures explicitly?

Spectral/projection-valued measures have very handy applications theoretically, but I got stuck when asked to compute explicitly certain projection-valued measures. Let's focus on the following: ...
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74 views

Approximating bounded operators in Hilbert space

Let $H$ be a separable Hilbert space, show that every bounded operator from H to itself can be approximated in the strong operator topology by a sequence of finite rank operators. I know we can find ...
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241 views

Graph of symmetric linear map is closed

A homework problem: Let $H$ be a Hilbert space. Let $T:H\rightarrow H$ be a symmetric linear map ($\langle Tx,y\rangle=\langle x,Ty\rangle$). Show that $S$ is bounded. My attempt: I'd ...
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58 views

Characterize compact sequences for a linear map.

Given a bounded sequence $\pi = (\lambda_n)$ in $\mathbb{C}$ consider the continuous linear map $M_\pi:\ell^2\rightarrow \ell^2$ defined by $$M_\pi(x_n) = (\pi_nx_n)$$ a) determine the spectrum. b) ...
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257 views

The Principle of Condensation of Singularities

Let $X$, $Y$ be Banach spaces and $\{T_{jk} : j,k \in\Bbb N\}$ be bounded linear maps from $X$ to $Y$. Suppose that for each $k$ there exists $x\in X$ such that $\sup\{\lVert T_{jk} x\rVert : j ...
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171 views

The convergence of the adjoint operator

If a sequence of operator $A_n$ converges in norm to $A$, i.e. $\lim \lVert A_n-A\rVert=0$)where $A_n$ and $A\in B(H)$ ($H$ is the Hilbert space). Is it true that $A_n^*$ converges in norm to $A^*$?
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192 views

Normal operators in Hilbert spaces

Let $H$ be a separable Hilbert space and let $T:H\to H$ be a continues linear map such that there exists an orthonormal basis of $H$ that consists of the eigenvectors of $T$. Show that $T$ is normal. ...
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165 views

Determine the operator T in a Hilbert space

Let $H$ be a Hilbert space and let $\{e_n, n \geq 1\}$ be an orthonormal basis for $H$. a) Determine the operator $T\in B(H)$ that satisfies $$ Te_1 = 0,\; Te_n = \frac{1}{n}e_{n-1}, n ...
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612 views

Projection operator in Hilbert spaces

Let T be a bounded operator on the Hilbert space H with the property that $T^*(T-I)= 0$. Show that T is an orthogonal projection. Im not really sure how to show that an operator is an orthogonal ...
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72 views

Convergence of a series in the $C^\infty$ topology

I am struggeling to understand the following argument given by F.Treves in the book "Introduction to Pseudodifferential Operators". The motivating problem for this is to find an approximate kernel ...
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93 views

Show that $(x_n)$ is in $\ell^2$

Let $x = (x_n)$ be a sequence of complex numbers with the property that for every $y = (y_n) \in \ell^2$ we have that the sequence $(S_N(y))_{N\geq1}$ with $$S_N(y) =\sum_{n=1}^N x_ny_n $$ converges. ...
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533 views

If $A_n$ is a sequence of positive bounded linear operators converging in norm to $A$ on a Hilbert Space, show $\sqrt{A_n}\to\sqrt{A}$ in norm.

If $A_n$ is a sequence of positive bounded linear operators converging in norm to $A$ on a Hilbert Space, show $\sqrt{A_n}\to\sqrt{A}$ in norm. I can show that $A$ would be positive and thus have a ...
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286 views

If a map $C:X\rightarrow U$ maps every weakly convergent sequence into strongly convergent

A Linear map between Banach spaces $C:X\rightarrow U$ is compact if it maps if the closure of the image of the unit ball is precompact in U. If a map $C:X\rightarrow U$ maps every weakly convergent ...
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94 views

Does $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for bounded operators on Hilbert space?

If $A$ is a bounded linear operator on a Hilbert space $H$ is it true that $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for all $x\in H$? If not, can we at least establish inequality in one ...
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106 views

Unbounded sets in infinite-dimensional normed spaces.

Let $X$ be an infinte-dimensional normed space. Let $\ell_1,\ldots, \ell_n$ be continuous linear functionals on $X$ and consider the set $$U = \{x\in X : |\ell_j(x)| < 1,\;\; 1\leq j \leq n\}.$$ ...
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73 views

Collection of linear functions

Let $X$ be a Banach space. Let $\{Y_\alpha\}_\alpha$ be normed spaces. Let $\{T_\alpha:X\rightarrow Y_\alpha\}_\alpha$ be an infinite collection of bounded linear functions. Is there a way to create ...
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1answer
473 views

Transpose of Volterra operator

I want to find the transpose of the Volterra operator $$Vf(x) = \int_0^x f(t)dt, \;\; x\in(0,1)$$ acting in $V:L^2(0,1) \rightarrow V:L^2(0,1) $. The transpose is defined as $\textbf{M}':U'\rightarrow ...
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80 views

Compactness of multiplication operator [duplicate]

Possible Duplicate: Compactness of Multiplication Operator on $L^2$ Let $u: \mathbb{R}\rightarrow \mathbb{C}$ be a bounded continuous function. Show that the multiplication operator $M_u$ ...
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240 views

Strong limit of compact operators

$X$ and $U$ are Banach spaces. A linear map $\textbf{C} : X \rightarrow U$ is called compact if the image $\textbf{C}B$ of the unit ball $B$ in X is precompact in $U$. A subset S of a complete metric ...
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201 views

$\sigma(x)$ has no hole in the algebra of polynomials

Let $A$ be a unital banach algebra generated by two elements $1$ and $x$. Then it seems $\sigma(x)$ cannot have holes. At least this is true in the case for disk algebras. This amounts to prove that ...
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54 views

Transpose of the Shift operators

Let $X = \ell^2$ The operators $\textbf{L}$ and $\textbf{R}$ are defined as $$\textbf{R}x = (0, a_0, a_1...) \;\; \textbf{L}x = (a_1, a_2, a_3...) $$ show that they are the transposes of one another ...
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Bounded functionals on Banach spaces.

Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], f\rightarrow f(r)$ defines a bounded linear functional on $X$. Prove that there exists a ...
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432 views

Bounded integral operators in Functional analysis

Let $K: [0,1] \times \mathbb{R}^n \to \mathbb{C}$ have the properties: $K(x,\cdot) \in L^2(\mathbb{R}^n)$ for all $x\in[0,1]$ For every $f\in L^2(\mathbb{R}^n)$ the function $$ x\mapsto ...
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256 views

(SOLVED) Adjoint of Frechet derivative (involving gradient operator)

I need some help with a problem (a homework/programming exercise) regarding the adjoint operator of the Frechet derivative of an operator. I have the forward operator $ F(a) = L_a ^{-1}f $ where ...
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298 views

Functional analysis summary

Anyone knows a good summary containing the most important definitions and theorems about functional analysis.
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96 views

Continuous operator on $L^\infty$

$1<p<\infty$ and $k\in L^\infty([0,1]^2)$ $(Tf)(s)=\int_{0}^{1}k(s,t)f(t)dt$ I want to show that it is a continuous operator $T:L^p([0,1]->L^p([0,1])$ Proof: What I need to show is that ...
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188 views

Open mapping theorem and second category

This seems like a fundamental result but I can not solve it of find an solution: Let $M:X\rightarrow U$ be a bounded linear map between Banach spaces. Show that if the range of M is a set of second ...
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70 views

Reference about Fredholm determinants

I am searching for a reference book on Fredholm determinants. I am mainly interested in applications to probability theory, where cumulative distribution functions of limit laws are expressed in terms ...
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50 views

Closed range for maps between banach spaces? [duplicate]

Possible Duplicate: Question about Fredholm operator This seems to be a standard result but I cannot find the solution. Let $M:X \rightarrow U$ be a bounded linear map between two Banach ...
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304 views

Show that the Volterra operator have dense range.

Let $V: C([0,1]) \rightarrow C([0,1])$ be defined by $$ V f(x) = \int\limits_0^x f(t) dt.$$ Show that V has dense range and find the transpose of V. V has dense range: Since the polynomials are dense ...
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31 views

Finding operator with specific properties

Let $H=(\mathbb R^2,(.,.))$ and $M=\{(x,0)|x\in\mathbb R\}, N=\{(x,x\tan(\theta)|x\in\mathbb R)$ with $\theta\in(0,\frac{\pi}{2})$. Now I would like to find a $T_\theta\in B(H,H)$ with ...
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Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
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237 views

Proof that certain operators are compact

I want to examine which of the following operators $T \colon C[0,1] \to C[0,1]$. are compact, by some I think I got the argument, but others I have no idea. a) $Tx(t) = x(t^2)$ Guess it is ...
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573 views

Neumann series and spectral radius

I have a question about the convergence of the Neumann series: Let $A$ be a matrix with spectral radius $\rho(A)<1$, i.e., all eigenvalues of $A$ are strictly less than $1$. Does that imply that ...
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Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...
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1answer
140 views

Inverse of trace class operator restricted to it's range

A paper I'm reading constructs the Cameron-Martin space in a way different than I'm used to, and in the process they gloss over a functional analysis result about the existence of an inverse. It ...
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743 views

Showing that the orthogonal projection in a Hilbert space is compact iff the subspace is finite dimensional

Suppose that we have a Hilbert Space $H$ and $M$ is a closed subspace of $H$. Let $T\colon H\rightarrow M$ be the orthogonal projection onto $M$. I have to show that $T$ is compact iff $M$ is finite ...
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760 views

Hilbert-Schmidt Operator

We have just covered Hilbert-Schmidt operators in class (which I missed) and I am having a hard time getting my head around them. I know the definition: If $H$ is a Hilbert space and ...
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1answer
227 views

Operation in Hilbert space with matrices

Let $\{e_n\mid n \in \mathbb{N}\}$ be an orthonormal basis for the Hilbert space $H$ and define for each $T \in B(H)$ the doubly infinite matrix $A = \{\alpha_{n,m}\}$ by letting $\alpha_{n,m} = (T ...
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587 views

Multiplication operator

Let $M_{\phi}$ be a multiplication operator $M_{\phi}:L^{2}\left(\mu\right)\rightarrow L^{2}\left(\mu\right)$ defined by $M_{\phi}f=\phi f$. Show that $\ker M_{\phi}=0$ if and only if ...
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156 views

Hilbert space the trace

I need help from someone to solve this problem. Given a bounded sequence $(\lambda_n)$ in $\mathbb С$ define an operator $S$ in $B(\ell_2)$ by $S(x_1) = 0$ and $S(x_n) = \lambda_n x_{n-1}$ , ...
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50 views

What is a Fourier decomposition of the index of an operator?

Consider a compact manifold $M$ equipped with some $S^1$-action, and let $E,F$ be vector bundles over $M$. Suppose further that a fixed elliptic operator $D:\Gamma(E)\to\Gamma(F)$ is preserved under ...
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228 views

Differential operators: elliptic vs strongly elliptic

This morning a collegue of mine came to me with the following question: does there exist any elliptic operator of order $2m$ with real (variable) coefficients that is not strongly elliptic? After ...
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437 views

Spectrum and point spectrum of this operator

Let $T\in \text{Aut}(\ell^2(\mathbb{C}))$ and $T(x)=(a_1 x_1, a_2 x_2,\ldots)$ where $a=(a_i)_i \in \ell^\infty(\mathbb{C})$. How can I easily see what is $\sigma(T)$ and $\sigma_p(T)$ (that are ...
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84 views

Computing $e^{isD}$ for a differential operator D

I'm trying to understand functional calculus of unbounded operators and everywhere I see proofs of its existence, but it seems that no one ever dares to compute some easy example. Lets take $D = ...
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232 views

operator norm of this multiplier operator

I am having some trouble with some basic properties of a given operator. Firstly, the operator T is defined as taking the fourier inverse transform of the function ...
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72 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
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570 views

Commuting operators and polar decomposition

Suppose that $V$ is an isometry and $X$ an arbitrary operator on a Hilbert space $H$. Let $X=U|X|$ be the polar decomposition for $X$. If $VX=XV$, can I conclude that $VU=UV$?