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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32 views

is this equality true?

Let $\{M_\alpha\}$ be a family of closed subspaces of a Hilbert space H. Is this equality true? ${(\cap M_\alpha^\bot)^\bot}=\overline{span(\cup M_\alpha)}$. thanks for your help.
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1answer
29 views

A question about state on tensor product of $C^\ast$-algebra

On the page78 of the book C*-algebras and Finite-dimensional Approximations there is a corollary as follows. Corollary 3.4.3. Let $\|\cdot\|_\alpha$ be a $C^\ast$-norm on $A\odot B$ and $\xi$ be a ...
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0answers
28 views

Prove operator convergence

I have to proove that given $X$ a normed space and $Y$ a Banach space, if the sequence of bounden linear operators from $X$ to $Y$ $\{A_n\} \rightarrow A$ and the sequence $\{x_n\} \rightarrow x$ then ...
4
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1answer
64 views

Compactness of set of projections

In a proof I am reading I saw the statement that the set of projections onto one fixed finite dimensional subspace of a Banach space $X$ is compact. Is this obvious, for some reason I just don't see ...
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3answers
50 views

Examples of spectrum of compact operators on the sequence space $l_2$

Suppose $T$ is a compact operator on the sequence space $l_2$, and let $\sigma(T)$ be its spectrum. Is it possible to find a $T \ne 0$ such that $\sigma(T) = \{0\}$? Also, is it possible to find $T$ ...
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1answer
24 views

Is the composite of a state and a $\ast$-homomorphism again a state?

If $\cal{A,B}$ are unital $C^\ast$-algebras and $\varphi:\cal{A\to B}$ is an unital $\ast$-homomorphism. Then it is clear that $\rho\circ\varphi$ is a state on $\cal{A}$ for any state $\rho$ on ...
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2answers
64 views

Sequence of unit vectors in a Hilbert space

Question: Let $H$ be a Hilbert space and $\{\xi_{i}\}\subset H$ be a sequence of unit vectors. Suppose that $||T_{j}(\xi_{i})-\xi_{i}||\rightarrow0$ as $i\rightarrow\infty$, for $j=1, 2, ...n$ (here ...
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0answers
32 views

Split Step Fourier Algorithm

Consider the NLSE (Nonlinear Schroedinger equation) that can be written as the following partial differential equation: $$ \frac{\partial{A}}{\partial{z}}=({\cal{L+N}})A\quad\quad(1) $$ where $A: ...
2
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1answer
49 views

An exercise about projections on Hilbert space

Let $H$ be a Hilbert space with an orthonormal basis $\{v_{n}\}_{n=1}^{\infty}$. The C$^{*}$-algebra $K$, the set of all compact operators on $H$, is a non-unital C$^{*}$-algebra. Let $p_{n}$ be the ...
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1answer
94 views

Show that $T$ is continuous

I have a question about how to response to: Given a Banach space X and $T: X \rightarrow X^{*}$ a linear operator such that $\langle Tx,x\rangle \geq 0$ for all $x \in X$. Show that $T$ is ...
2
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1answer
78 views

Maximal monotone operator without convex domain?

I'm looking for an example of a (multi-valued) maximal monotone operator $A$ mapping a Banach space $X$ into its dual $X^*$ such that the domain $D(A)=\{x\in X: Ax\neq\emptyset\}$ is not convex. ...
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1answer
45 views

Multiplication operator has no no eigenvalues

Statement: Let $M_x$ denote the multiplicative operator acting on $L^2([0,1], \, dx)$ by $M_x(f) = xf$. Show that $M_x$ has no eigenvalues Attempt: Let $M_x(f) = xf$ then we should have $M_x(f) = ...
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0answers
25 views

Second Level Operators:

What would be an example of an Operator $$H$$ such that for any and all explicit functions U $$H[u] = I$$ where I is some other function However, for some other Operator W ex: [d/dx] ...
4
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1answer
41 views

Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
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0answers
19 views

about spectrum of the sum of two operators

It is well known that if $A$ and $B$ are commuting bounded operators on a Banach space then it is familiar that $$ \sigma(A+B)\subseteq\sigma(A)+\sigma(B)$$ I would to ask if I can find suffisant ...
2
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1answer
71 views

Convergence Radius => Nonanalytic

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
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1answer
21 views

A question about states for nonunital $C^\ast$-algebras

Let $\rho$ be a state on a nonunital $C^\ast$-algebra $\cal{A}$, $a\in\cal{A}$ and $\rho(a^\ast a)\neq0$. Define $\rho_a:b\mapsto\rho(a^\ast ba)/\rho(a^\ast a)$. Then $\rho_a$ is a state on $\cal{A}$. ...
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1answer
21 views

stability of essential spectra

Let $X$ be a Banach space. $A$ and $B$ are linear closed and densely defined operators and $\lambda\in\rho(A)\cap\rho(B)$ such that $(\lambda - A)^{-1}-(\lambda - B)^{-1}$ is a Frehholm perturbation ...
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1answer
29 views

Is the composition of monotone operators monotone?

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
10
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2answers
239 views

Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } ...
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0answers
36 views

$K(X,Y)$ is closed in $B(X,Y)$

I tried to prove that the set of compact operators $K(X,Y)$ is a closed subset of the set of bounded operators $B(X,Y)$ where $X,Y$ are Banach spaces. Please can you tell me if my proof is correct? ...
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0answers
25 views

Show that $B(X)$ is semisimple for a Banach space $X$ [duplicate]

Show that $B(X)$ is a semisimple Banach algebra, where $X$ is a Banach space. That is, to show that rad $B(X)=\{0\}$, or equivalently, to show $\sigma(AT)={0} \, \forall T\in B(X)\Rightarrow A=0$. I ...
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1answer
55 views

Exponential of matrices and bounded operators

Let $A$ be a complex $n \times n$ matrix, such that the function $t\mapsto e^{tA}x$ is bounded on $\mathbb{R}$ and nonzero, for some vector $x\in \mathbb{C}$. How can we prove that $\inf_{t\in ...
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1answer
78 views

Question about Hahn-Banach theorem

Let $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ be normed spaces, and $X\subset Y$. If each $f\in (X,\|\cdot\|_1)^\ast$ extends to a bounded linear functional in $(Y,\|\cdot\|_2)^\ast$ with same norm, ...
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2answers
81 views

Compact operator on $L^{2}$

Let $K(t,s)$ be a real-valued function of two real variables, and let $T: L^2(a,b) \to L^2(a,b)$ be defined by $(Tf)(t) = \int_{a}^{b} K(t,s) f(s) ds$ where $$K(t,s) = \sum_{j=1}^{n} \phi_{j}(t) ...
2
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1answer
133 views

Compact and bounded if and only if $X$ is finite dimensional

I tried to prove the theorem Let $X$ be a Banach space. Then $K(X) = B(X)$ if and only if $X$ is finite-dimensional. Please could someone check my proof? (The implication where $X$ is finite ...
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1answer
47 views

Show compactness of an operator with Arzelà–Ascoli

We have $K\colon L^{2}(a,b) \rightarrow L^{2}(a,b)$ such that $ Kf(t)=\sum_{j=1}^{n}\phi_{j}(t) \int_{a}^{b} \psi_{j}(S) f(s)ds$ where $\phi_{j} ,\psi_{j} \in L^{2}(a,b)$. We want to show that K is ...
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0answers
94 views

Compact operators is a linear subspace of bounded operators

Let $X,Y$ be Banach spaces. Let $B(X,Y)$ be the set of bounded linear operators and let $K(X,Y)$ be the set of compact linear operators. I want to prove that $K(X,Y)$ is a vector subspace of ...
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1answer
106 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
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1answer
26 views

Check continuity of linear functionals and find norms

1) $c_{00} \owns (x_n) \mapsto \sum_{n=0}^{\infty} x_n \in \mathbb{K}$ where $c_{00}$ is a space of sequences that are eventually equal to $0$ with sup norm 2) $\ell^\infty \owns (x_n) \mapsto ...
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1answer
112 views

norm of integral operator in $C([0,1])$

If we define on $C([0,1])$ the operator $$ Tf(x) = \int_{0}^{1} K(t,s) f(s) ds$$ where $K$ is a continous function on two variables. I want to show that: $1)$ $||T|| = \displaystyle\max_{t} ...
2
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2answers
131 views

Proof of equivalent characterizations of compact operators

As an exercise I tried to prove the following theorem: If $X,Y$ are Banach spaces and $u \in B(X,Y)$ is a bounded linear operator then the following are equivalent: (1) $u$ is compact ...
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0answers
139 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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1answer
42 views

How do I show a left inverse of a bounded linear operator on Banach space?

If $A$ is a bounded linear operator on a Banach space X, with a left inverse $A_l^{-1}$, and P is a projection (also on X), how do I show that $A_l^{-1}P$ is also a left inverse of A (i.e. ...
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0answers
68 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
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0answers
25 views

Limit of exp of self-adjoint operator

Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator ...
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1answer
42 views

Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
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0answers
37 views

Pseudo-monotone operators research paper question

Hi I just want to know if anyone can see how the result (2.34) is obtained in the following research paper http://caa.epfl.ch/publications/9-Boccardo-Dacorogna1984.pdf. Thanks, I know that it is a ...
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2answers
36 views

Bijective Bounded Operator Extension: Where do the new elements go to?

Given a dense, proper subset of complete spaces: $$X,Y\text{ both complete and }A\subsetneq\overline{A}=X$$ and an operator between them: $$T:A\to Y\text{ continuous, linear and bijective}$$ Now, ...
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2answers
78 views

Isometry <=> Adjoint left inverse [duplicate]

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
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2answers
71 views

Isometric <=> Left Inverse Adjoint

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
2
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1answer
77 views

Norm of a projection in $L_p$

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is ...
2
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1answer
72 views

is this true for Hilbert space direct sum of $H$ when $H$ is infinite dimensional?

Let $(H_{\alpha})_{{\alpha \in I}}$ be a $I-$indexed family of Hilbert spaces over $\mathbb{F}$. let $H=\bigoplus H_\alpha$ be their Hilbert space direct sum. Can we say $\dim ...
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1answer
22 views

Common eigenvector of a sequence of compact operators

Let $H$ be a separable, infinite-dimensional Hilbert space and suppose we have a sequence of norm-one compact operators $(A_n)$ on $H$ which all have 1 as an eigenvalue. Can we pass to a subsequence ...
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2answers
40 views

Immediate consequence of the definition of Operator Norm. Explain

||Av|| $\leq$ ||A||$_{op}$||v|| for every v $\in$ V I was wondering why this is true. Wikipedia says it's an immediate consequence of the definition but I just do not follow. I am using the ...
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1answer
19 views

The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...
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1answer
27 views

Positivity of certain matrix

Let $A=[[a_{ij}]]$ and $B=[[b_{ij}]]$ be two positive semi-definite matrices of same dimensions. Further they have a property that, if $a_{ij}=0$ then $b_{ij}=0$ (i.e. the nonzero entries appear in ...
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0answers
27 views

Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is ...
2
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2answers
114 views

Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
1
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1answer
22 views

Integral's limit

Let $X$ be a Banach space and $A$ is a linear bounded operator on $X$. It is well known that for $|\lambda|> \|A\|,$ we have $$\|(\lambda I - A)^{-1}\| \leq \frac{1}{|\lambda|-\|A\|}.$$ Now, let ...