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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Unital free semigroup and operators

Let $F_n^+$ be a unital free semigroup generated by $1,...,n$. Let $\alpha= i_1...i_k$ where $i_1,...,i_k\in \{1,...,n\}$ and put $T_{\alpha}:=T_{i_1}...T_{i_k}$ where for any $ i_j\in \{1,...,n\},~ ...
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Spectral theory and sequences: is this fact a general truth or does it depend on the operator?

Let $\lambda\in\mathbb{R}\setminus\{0\}$, $\textbf{i}$ the imaginary unit, $H$ a Hilbert space, $T:D(T)\subset H\to H$ a invertible densely defined linear operator such that $T^{-1}$ is bounded, ...
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Show these operators converge to a particular limit

Let $H$ be a Hilbert space, and $T$ be a operator on $H$ of the form $T=\sum_{n=1}^{\infty}{\lambda}_{n}<x,e_{n}>e_{n}$ where $e_{n}$ are the eigenvectors of $T$ and an orthonormal basis of H ...
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Comparison of operators

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any ...
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An exercise in operator theory

Let $H$ be a Hilbert space and $P$ be a projection to a finite dimensional subspace $K$ of $H$, for a $T\in B(H)$, if $||PTP||=1$, then, for arbitrary $\epsilon>0$, there exists a vector $\alpha$ ...
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55 views

Invariant convex subset

Let X be a banach space and $T$ a bounded operator on X with $||T||$ less than or equal to 1. If $T$ is an isometry and $r(T)$<1 show that there is a closed convex non-zero proper subset $C$ of the ...
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Powers of a closed range operator

suppose that $S$ and $S^2$ are operators with closed range. Does it follow that $S^n$ is an operator with closed range for all natural numbers $n$?
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28 views

Kernel closed implies continuous operator?

Is closed kernel sufficient for linear operators to be continuous? Counterexample? Thx, Alex
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Compact operators and dimension

If every compact operator is of finite dimensional range,does it follow that the dimension of the space is finite?
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If $u_m \rightharpoonup u$, how to show using monotonicity that $f(u_m) \rightharpoonup f(u)$?

Let $$u_m \rightharpoonup u \quad \text{(weakly) in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$}.$$ We are given $f:\mathbb R \to \mathbb R$, a Lipschitz continuous invertible map which is ...
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Rotating the spectrum of a bounded operator

If $T$ is a bounded operator on a Banach space $X$, and $\sigma(T)$ is its spectrum, what would be an operator whose spectrum is $\sigma(T)$ rotated by $\theta$? For example, $-T$ has as spectrum ...
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composition and commutators of Fourier multiplier operators

I am working with some Fourier multiplier operators arising in study of a PDE. I have a general question: Suppose $S$ and $T$ be two Fourier multiplier operators (on some space) with multipliers $m_1$ ...
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Fuglede's theorem

Let $T$ be a normal operator and let $A$ commute with $T$. Then $A$ commutes with the $T^*$. I'm having a few difficulties namely: 1) Show that $||e^{itB}||$ is less than or equal to 1 where $B$ is ...
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Range of the product of two operators

Is the Range of the product of two operators with closed range have closed range? What extra conditions must we impose?
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Distributions with a given mean and covariance

Fix $X := \mathbb R^d$ for some $d \ge 1$. Fix a vector $m \in X$ and a covariance operator $k : X^* \to X$, i.e., a symmetric, nonnegative-definite operator. Let $\Delta_{m,k}(X)$ denote the ...
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Bounded operators with closed range

Under what conditions the sum of two closed range operators have a closed range? the result is true if the ranges are orthogonal(hilbert space)
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spectral measure of non-empty and open set is non-zero proof

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf Part d) on page 322 and his proof appears on page 324. I didn't quite understand his proof so I had a go at ...
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Properties of solutions to an ODE

I have an ODE: $$ \frac{\mathrm{d}u}{\mathrm{d}t} + \mathcal{A}(t, u) = 0 $$ with final condition: $$ u(T)= \mathbf{1} $$ The function $u:\mathbb{R} \rightarrow \mathbb{R}^m$ is vectorial, and the ...
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59 views

Random operators

Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H ...
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36 views

invariant subspace partition

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf on page 327 Rudin says that M and M' are invariant subspaces. I'm guessing he means non-trivial so how does ...
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Proving an operator $D: L^2[0,1]\rightarrow C'$, $Df(t)=\int^t_0 f(s) ds$ is unitary

Let $C'\subseteq C[0,1]$ be the space of all absolutely continuous function such that $f(0)=0$ and $f' \in L^2[0,1]$. Define an inner product on $C'$ as $\langle f,g \rangle = \int^1_0 f'(t)g'(t)dt$. ...
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Abstract Wiener space and integration related to a trace class operator

Suppose I have a trace class operator $A$ of a Hilbert space $H$. Also suppose I have an abstract Wiener space $(H,B)$. Then, $\langle Ax, x \rangle$ is defined almost everywhere in $B$ with respect ...
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Rudin's proof of invariant subspace existence

I have questions about Rudin's proof of invariant subspace existence. On page 327, point 12.27, How does he get that $Tx=TE(\omega)x$, and How does he know $E(\omega)$ is not the zero map? ...
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Random operators [duplicate]

Let $(\Omega, \mathcal F,P)$ be a probability spaces and $H$ be a Hilbert space. By a random operator $A$ from $H$ to $H$ we mean a linear continuous mapping from $H$ into the Frechet space $L_0^H ...
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Subspaces in the image of compact operator

Let $X$ and $Y$ be some infinite dimensional Banach spaces. Let $T:X\longrightarrow Y$ be some compact linear operator. It is easy to understand that $T$ cannot be surjective: the Open Mapping Theorem ...
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Rudin functional analysis problem

-rudin-2th.pdf">http://59clc.files.wordpress.com/2012/08/functional-analysis--rudin-2th.pdf 1) page 326, he says that if ST=TS, then S commutes with f(T). He has previously shows that if S commutes ...
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Find a real matrix with eigen vectors v and v's complex conjugate so that they have different eigenvalues.

I need to find a real matrix with eigenvector v, and eigenvector v's complex conjugate, such that they will have different eigenvalue. any hints please?
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Proof that the Sturm-Liouville operator is positive

Define the Sturm-Liouville operator $L$ on $[a,b]$ such that $Lu = \frac{1}{\omega}[-(pu')'+qu]$, $p$ and $\omega$ being strictly positive real valued function, and $q$ a positive real valued ...
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66 views

Normal Self-Invertible Operator is Self-Adjoint

If $T\in B(H)$ for some Hilbert space $H$, is a normal operator and $T^2=I$, then $T=T^*$. It seemed simple when I first saw the claim, but I'm having trouble showing it. I know that it implies ...
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A question about $C^\ast$-algebra

In Kadinson's book Fundamentals of The Theory of Operator Algebra, when the author proved the Theorem 7.2.1, he let $V$ be an extreme point of the unit ball of $C^\ast$-algebra $\cal{U}$, $h$ be a ...
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Toeplitz operators

i'm writing here to ask an explanation about Toeplitz operators. I'm trying to understand how they are defined and them basic properties. Thank you in advance for your consideration.
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Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
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Examples of spectral decompositions

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, ...
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51 views

Infinite dimensional operator inverse

A is a linear operator on V and there exist a single operator B on V such that AB = I or BA = I. Prove that then A is monomorfic and epimorfic. On infinite dimensions, left and right inverses need ...
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Totally continuous implies bounded

Consider a separable, reflexive Banach space $V$. We define the mapping $A: V \rightarrow V^{*}$ as totally continuous if it is continuous as a mapping $(V, \text{weak}) \rightarrow (V^{*}, norm)$. I ...
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A boundedness of identity operator

Suppose $I\colon L_p(0,1)\to L_q(0,1),\,1\leq p < q <\infty$. I claim that this operator even is not defined over all $L_p(0,1)$. Therefore, we cannot speak about its boundedness. Am I right?
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The unitary implementation of $*$-isomorphism of $B(H)$

Is it possible to construct $*$-isomorphism of (factor von Neumann) algebra $B(H)$ which is not unitary implementable?
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equality of spectrums from 2 lemmas

Let $A$ be a $B^{*}$ algebra and let $B$ be a sub $B^{*}$ algebra. Deduce from the following 2 lemmas that for $x$ in $B$ we $x$ is invertible in $B$ iff it is invertible in $A$. Lemma 1) Let $x$ in ...
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Multiplication operator is not jointly continuous in strong topology

How can I show that multiplication operator ($M:\mathcal{L}(X,Y) \times \mathcal{L}(Y,Z) \rightarrow \mathcal{L}(X,Z)$; $M(A,B)=AB)$ is not jointly continuous in strong topology? I have to show that ...
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Alternative definition of strong/weak operator topology.

Given two normed spaces $(X, ||\cdot||_X)$ and $(Y,||\cdot||_Y)$ the space of bounded linear maps $\mathcal{B}(X,Y)$ can be equipped with the strong operator topology (SOT) as follows: The ...
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Perturbation of eigenvalues

I am looking at a certain operator, that is an integral operator which is Hilbert-Schmidt on $L^2(X,d\mu)$ to $L^2(X,d\mu)$.I want to see how its eigenvalues or singular values change as its kernel is ...
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Multiplication operator is closed

Let $\Omega$ be an unbounded subset of $\mathbb{C}$. Consider the Hilbert space $H=L^{2}(\Omega)$ and the (unbounded) multiplication operator $T$ given as follows: $Tf(z)=zf(z), f(z)\in D(T)$ where ...
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Automorphism of $W^*$ algebra

Let $\mathfrak{A}$ be von Neumann algebra. It is in particular $C^*$ algebra. Is it true that every $*$-isomorphism of $\mathfrak{A}$ is also $W^*-$isomorphism? (Note that every $*$-isomorphism of ...
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Constants in Maximal regularity

We consider the following evolution equation \begin{equation} \left\{ \begin{array}{llc} v_t=A v+f,\\ v(0)=0. \end{array} \right. \end{equation} $A$ generates a bounded analytic semigroup on a Banach ...
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Non-empty closed subset of the complex plane is the spectrum of a normal operator

This is an exercise in Chapter 13 of Rudin's Functional Analysis. The question is to show that every non-empty closed subset of $\mathbb{C}$ is the spectrum of some normal (not bounded) operator in a ...
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Linear Operator identity prrof [closed]

Let A,B be invertible linear operators. Prove the identity: $B^{-1}-A^{-1}=B^{-1}(A-B)A^{-1}$
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21 views

A separating set which is not cyclic

Let $H=L^2[0,1]$ , $T_g$ be the multiplication operator on $H$, i.e. $f\to fg$ . Let $A$ be the set of the $T_g$ as $g$ runs through the set of polynomials with complex coefficients. Let $h$ be te ...
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Strong and weak equivalence of $C^*$-extensions by compacts

Let $A$ be a $C^*$-algebra. An extension of $A$ by the compact operators $K$ is an embedding $\epsilon$ of $A$ into the Calkin algebra $B(H)/K$. Two embeddings $\epsilon_1$ and $\epsilon_2$ are ...
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Inequality on Hilbert spaces in order to prove the nonexpansivity of a mapping.

I have an application $T\colon H\to H$ (where $H$ is a Hilbert space) such that $$(Tx-Ty,x-y)\leq \|x-y\|^2,\forall x,y\in H$$ where $(\cdot,\cdot)$ is the inner product of $H$ and $\|\cdot\|$ its ...
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Hydrogenhamiltonian self-adjoint in one or two dimensions

let $d\in\{1,2\}$. I'd like to know if the operator $H=-\Delta - \frac{1}{|x|}$ is self-adjoint as an operator acting on a dense subset of $L^2(\mathbb R^d)$. In particular I'd like to know how its ...