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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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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13 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 ...
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54 views
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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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$K(X,Y)$ 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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19 views

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

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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28 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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64 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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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) ...
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52 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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22 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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59 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
36 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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20 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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55 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} ...
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66 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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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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19 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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16 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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21 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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37 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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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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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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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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44 views

Isometry: Adjoint = Leftinverse

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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48 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 ...
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53 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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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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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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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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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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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 ...
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Why does exponentiating the derivative yield the shift operator?

If we formally exponentiate the derivative operator $\frac{d}{dx}$ on $\mathbb{R}$, we get $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
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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 ...
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The image of an bounded and linear operator between two banach spaces is closed if the image is finite codimension?

So here is my problem, I would like to prove the following, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded operator. Then $TX$ is closed if it is of finite codimension i.e ...
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How to get a grip on codimensions

I am trying to find a proof for the following problem: Let $X,Y$ be Banach spaces $A,B:X \rightarrow Y$ are bounded linear operators and $Ran(A)$ is closed. If $\left \| Bx\right \|<\left \| ...
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1answer
22 views

Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
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Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
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96 views

Spectral properties in Hilbert Space

If $X$ is a Hilbert space, then prove that $$σ(T^*)=\overline{σ(T)}\hspace{2cm}\text{and}$$ $$[R_\lambda(T)]^*=[R_\overline{\lambda}(T)]\hspace{1cm}\text{for all $\lambda\in\rho(T)$ ,}$$ where ...
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Show that any compact set in $\mathbb{C}$ is the spectrum of an operator.

I have been looking around for an example of a general continuous (bounded) linear operator, who's spectrum is any compact set $K\subset\mathbb{C}$. I have seen an example, where we take the set ...
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Confusion with proving that some subspace of a Banach-Space is closed

So here is my problem, I am trying to show that, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension. While ...
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Equivalent conditions for composition to be compact operator

I did some exercises in Conway's functional analysis book and found the following problem: Let $\tau:[0,1]\to [0,1]$ be continuous and define $A:C[0,1]\to C[0,1]$ by $Af:= f\circ \tau$. Give ...
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1answer
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Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
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23 views

Smoothness of solutions to Fredholm integral equation

Let $K(x,y)=k(|x-y|)$ where $k$ is continuous on $(0,1]$, and assume function $f\in L^2[0,1]$ satisfies $f(x)=\int_0^1 f(y)K(x,y)dy$. Is $f$ necessarily $C^\infty $ ? under what condition on kernel ...
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Perturbation of resolvent

Let $H$ be a Hilbert space.$A$ is densely defined closed operator and $B$ is $A$-bounded:$||Bx||\leq a||Ax||+b||x||$.Let $\lambda\in\rho(A)$,such that ...
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Solving the equation $z=\sum_i \alpha_i \exp(-\|x_i-z\|^2)x_i$

For $i=1,\dots,M$ vectors $x_i\in\mathbb{R}^N$ and scalars $\alpha_i$, can you find a vector $z$ satisfying the equation $z=\sum_i \alpha_i \exp(-\|x_i-z\|^2)x_i$? Any pointers will also be ...
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Images of unitaries

Let $n\geqslant 0$. Suppose that $U$ is a unitary matrix in $M_n$ and there are two unital ${}^\ast$-homomorhpisms $\pi_1\colon M_n\to A, \pi_2\colon M_n \to B$, where $A,B$ are C*-algebras such that ...
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49 views

Question about the essential spectrum of a negative difinite operator

please on an infinite dimensional Hilbert space how to difine the essential spectrum of an operator which is negative definite ??? Please help me Thank you.
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Domain of the quantum free Hamiltonian in 1D

Consider the quantum free Hamiltonian $H_0 =-\frac{d^2}{dx^2}$ (the Laplacian on the real line). I want to show that it is (essentially) self-adjoint in its domain of definition. The usual approach ...
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14 views

Orthogonality Condition Eigen Functions of Sturm-Liouville Operator

I was wondering if anyone could help to derive the orthogonality condition $$\int^b_a y_n(x)y_m(x)w(x)dx=\delta_{nm}$$ of the normalised eigenfunctions (denoted by $y_n$ with eigenvalues $\lambda_n$) ...
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H P S class operators and their inequalities

First few definitions: $A \in I(K)$ iff $A$ is isomorphic to some member of $K$ $A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some ...