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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Operator defined via a sequence of weights

Let the linear operator $T:l^2\rightarrow l^2$ be defined by $y=Tx$ where $x=\{\xi_j\}$, $y=\{\eta_j\}$, and $\eta_j = \alpha_j \xi_j$, where $\{\alpha_j\}$ is a dense sequence in $[0,1]$. Does ...
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328 views

Every Hilbert-Schmidt is an integral operator?

Let $(X,\mu)$ be a $\sigma$-finite measure space. If $K\in\mathcal{L}^2(X\times X,\mu\times\mu)$ then the map $A_K:\mathcal{L}^2(X,\mu)\to\mathcal{L}^2(X,\mu)$ defined by\begin{equation} ...
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Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
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303 views

cyclic vector exists for symmetric operator iff there no repeated eigenvalues

Considering a symmetric operator $A$ acting on a finite dimensional Hilbert space $H$, we say $x\in H$ is a cyclic vector for $A$ if the set of finite linear combinations of $\{A^n x:n=0,1,2,...\}$ is ...
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3answers
267 views

Compact integral and multiplication operator in Banach spaces

Let $ A\colon C[0,1] \to C[0,1] $ $$ A(x)(t) = f(t)x(t) + \int_0^t x(s)ds,\quad f \in C[0,1]: f(1) \neq 0, \forall t \in [0,1] $$ Is $A$ a compact operator or not?
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212 views

Operators with eigenvalue $\{0,1\}$ that is not projection

Show that there are linear operators T on the Hilbert space H what are not orthogonal projections, but their spectrum consists of the eigenvalues $\{0,1\}.$ I can not come up with an counterexample, ...
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77 views

Power series of bounded linear maps

Given a Banach space $X$ and a bounded linear map $T:X\rightarrow X$ we define $$e^T = I + \sum_{n\geq1}\frac{T^n}{n!}$$ Show that if $e^T$ is compact then dim $X<\infty$. I have showed before ...
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68 views

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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383 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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157 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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62 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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191 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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1answer
57 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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217 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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1answer
158 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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177 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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156 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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2answers
500 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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68 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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90 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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412 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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208 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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92 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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94 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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1answer
71 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
341 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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73 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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1answer
192 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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190 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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51 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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1answer
328 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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209 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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239 views

Functional analysis summary

Anyone knows a good summary containing the most important definitions and theorems about functional analysis.
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94 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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1answer
157 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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1answer
60 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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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 ...
3
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1answer
260 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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30 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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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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453 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
114 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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3answers
603 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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606 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
187 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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462 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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1answer
153 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 $S(x_n) = \lambda_n x_{n-1}$ , ...