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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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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17 views

Isometry <=> Adjoint left inverse

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse to }T$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle ...
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Isometry: Adjoint = Leftinverse

Given an isometric operator is it true that its adjoint is necessarily leftinverse? My attempt goes like this: $$\langle x,\mathbb{1}\tilde{x}\rangle=\langle x,\tilde{x}\rangle=\langle ...
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21 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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47 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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17 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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18 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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13 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
26 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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12 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 ...
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55 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 $$e^\frac{d}{dx} = I+\frac{d}{dx}+\frac{1}{2!}\frac{d^2}{dx^2}+\frac{1}{3!}\frac{d^3}{dx^3}+ \cdots$$ ...
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20 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 ...
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1answer
29 views

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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61 views

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
20 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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94 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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1answer
48 views

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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18 views

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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32 views

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
12 views

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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27 views

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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1answer
19 views

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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1answer
47 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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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 ...
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1answer
47 views

Inequality of Class operators H S and P

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 ...
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About locally convex space

Is a Banach space a locally convex space? Why? Recall A locally convex space is a linear topological space in which the topology has a base consisting of convex sets.
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About inverse of an operator

Let $ X_{p}:= L_{p}([-a,a]\times[-1,1], dxdv), (a>0,\, 1\leq p<\infty)$ and the operator $$\left\lbrace \begin{array}{l} S :X_{p} \rightarrow X_{p} \\ \qquad \psi \mapsto ...
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1answer
41 views

A question about utilizing Hahn-Banach theorem

There is a quotation below: Let $A$ be a Banach space, $\mathbb{B}(A)$ be the space of all bounded linear maps from $A$ to $A$ and $C\subset \mathbb{B}(A)$ be any convex set. If a net ...
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Domains of operators defined by quadratic forms

Consider a separable Hilbert space $H$. Say we have two lower-bounded, densely defined quadratic forms $a$ and $b$ with respective domains $D[a],D[b] \subset H$ such that $D[b] \subset D[a]$ ...
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Space of operators on function

Consider the following space of operators on function of $n$-variables $A= Span \{x_ix_j\ , x_i \frac{\partial}{\partial x_j} , \frac{\partial^2}{\partial x_i \partial x_j} , i,j=1,2,\cdots,n\}$. ...
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1answer
36 views

Composing Projections on a Hilbert Space

Let $P,Q$ be projections on a Hilbert space such that $PQ$ is a projection. I have been able to prove that $PQ=QP$. I want to show that $ker(PQ)$ is contained in $ker(P)+ker(Q)$. If there's a ...
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2answers
32 views

An approximation question on projections

Suppose $\{p_i\}_{i=1}^{m}$ are projections in the d by d matrix algebra $A$ over the complex numbers and satisfy the following condition: $||Id-\sum_{i=1}^m{p_i}||_2<c$, $||p_ip_j||_2<c, ...
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46 views

Self adjointness for functionals

I have posted this question already in the physics forum, but actually nobody could help. I am sorry, this question is related to quantum field theory. The Schrödinger equation of a free scalar field ...
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1answer
30 views

Composition property of the functional calculus

Prove from the spectral theorem for normal operators $T$ that for bounded borel functions $f,g$ we have $f{\circ}g(T)$=$f(g(T))$.
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69 views

Proving this operator is not closed

I was looking for an example of an operator on a Hilbert space which is not closed. This is what I have done so far, but am not sure if the proof is alright. Consider $H=L^{2}[0,1]$ and $T$ defined ...
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Why $ \|x^* x \| = \|x\|\|x^*\|$ is equivalent to $\|xx^*\| = \|x\|^2$ in the definition of $C^*$ algebra?

I read the definition of $C^*$ algebra in Wikipedia where it says $\|x^* x \| = \|x\|\|x^*\|$ is equivalent to $\|xx^*\| = \|x\|^2$ but I do not know why. Can you show me how to derive $\|xx^*\| = ...
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Transforms with $O(N \log N)$ Complexity

Beside the Discrete Fourier and Walsh-Hadamard operators, are there any non-trivial, bijective operators that admit an evaluation algorithm of $O(N \log N)$ time complexity or better, whose inverses ...
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1answer
33 views

Domain of adjoint of operator (example from Reed-Simon).

I am trying to understand this example from Reed-Simon volume 1: "Suppose that $f$ is a bounded measurable function, but that $f\notin L^2(\mathbb R)$. Let $D(T) = \{\psi \in L^2(\mathbb R) : ...
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49 views

polar decomposition proof

Let $H$ be a hilbert space and $T$ a bounded linear operator on $H$. I'm trying to prove that there is a partial isometry $V$ on the closure of $Im(|T|)$ such that $T=V|T|$ and $|T|=V^*T$, where ...
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1answer
27 views

What does the adjoint operator do? Is this Frechet derivative correct?

Problem statement Let $x \in l^2$ and $J(x) = \sum_{n = 1}^{+\infty} x_{2n - 1}^2$ Find first and second Frechet derivatives. Attempted solution Let's note that $J(x) = \sum_{n = ...
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1answer
51 views

show that the operator $T:l^2\rightarrow V$ is bounded

Let $V$ be the Banach space of all sequences $v=\{\eta_j\}_{j=1}^\infty$ such that $\lim_{j\rightarrow\infty}\eta_j$ exists. The norm on $V$ is given by $\|v\|=\sup_{j\in \mathbb{N}}\eta_j$. Consider ...
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57 views

uniqueness of positive operator

Let $A,B$ be commuting positive operators on a hilbert space such that $\langle(A-B)(A+B)x,x\rangle=0$ for all $x$ in the hilbert space. Prove that $A=B$. My attempt: The above implies that $A=B$ on ...
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1answer
80 views

$\langle Tx,x \rangle=0$ proof

If $T$ is a bounded operator on a hilbert space $H$ and $\langle Tx,x \rangle=0$ for all $x$ in $H$, then $T=0$. I'm considering what we can conclude if $\langle Tx,x \rangle=0$ for all $x$ in some ...
2
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1answer
26 views

Regularity estimates for parametrized family of elliptic operators

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Suppose we have a parametrized family of linear operators $\{A_\varepsilon\ :\ \varepsilon\in\mathbb{R}_{\geq0}\}$ such that ...
3
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1answer
42 views

An exercise of positive element in C*-algebra

Let $A$ be a unital C*-algebra and $\{b_{n}\}$ be a positive invertible sequence in $A$. If $||1_{A}-b_{n}||\rightarrow 0$, can we conclude $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$ ?