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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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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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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115 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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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 ...
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57 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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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 $Ran(A)$ is closed, and $\dim(\mathrm{Ker}(A))$ or ...
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28 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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23 views

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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75 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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25 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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1answer
48 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
28 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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27 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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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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21 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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65 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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52 views

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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58 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 ...
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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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51 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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40 views

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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55 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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55 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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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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82 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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50 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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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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39 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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65 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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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
113 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 ...
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35 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 ...
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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 $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$ ?
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Lack of a polar decomposition

Prove that the left and right shifts on $l_{2}$ have no polar decomposition (i.e. $UP$ where $U$ is unitary and $P$ is positive).
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Operator Norm = 1

Let there be a linear map $T$ such that $T: \mathbb R^n\to\mathbb R^m$. The operator norm $\lVert \cdot\lVert_{op}$ of $T$ is then defined as the largest value of $c$ for which $\lVert T(\vec v)\lVert ...
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136 views

square root of positive operators

It $T, S$ are positive operators, do we have that $\sqrt{TS}=\sqrt{T}\sqrt{S}$? Are there any basic rules that hold for square roots of positive numbers that don't hold for positive operators?
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50 views

The spectrum of the operators

Let $X, Y$ be the Banach space, and $T_{1}: X\rightarrow X$ and $T_{2}: Y\rightarrow Y$ be the bounded linear operators. Then what is the relationship between $\sigma(T_{1})$, $\sigma(T_{2})$ and ...
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76 views

functional calculus and spectral measure

Let $T$ be a normal operator and $f$ be a bounded borel function on ${\sigma}(T)$. If $E_{T}$ and $E_{f(T)}$ are the spectral decompositions of $T$ and $f(T)$ respectively, prove that for any borel ...
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Operator's comparison

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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257 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
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93 views

generalized eigenspace direct sum

Similar to the way an infinite dimensional hilbert space can be written as a direct sum of eigenspaces of a normal compact operator, I was wondering whether it can be written as a direct sum of ...
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39 views

The norm of operator matrix

Let $H$ be a Hilbert space and $B(H)$ be the bounded linear operator on $H$, for $T\in B(H)$, if $T=\left(\begin{array}{ccc} 0 & B \\ A & 0 \\ \end{array}\right)$ on $H=M\oplus ...
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1answer
37 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...