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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Metrics from Operator Norms

Let $X$ be a Hilbert space and $(\cdot,\cdot)_X$ be the inner product on $X$. It is well known that $|x|_X = \sqrt{(x,x)_X}$ is a norm on $X$ and $|x-y|_X$ is a metric on $X$. The norm on $X$ induces ...
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Norm of operator $A$ st. $A^2 = I$?

I'm wondering what can be said about the norm $||A||$ of an operator which squares to identity. All I can think of is that $$1=||AA|| \leq ||A||^2$$ so that $||A|| \geq 1$. But can anything else be ...
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spectral projection of an isolated point in the spectrum of a closed linear operator

Suppose that $T$ is a closed densely defined operator on a Hilbert space $H$ with $\rho(T) \neq \emptyset$. If $\lambda \in \sigma(T)$ is an isolated point then we know that $H = \mathcal{N}(E_o) \...
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Weak convergence and strong convergence on $B(H)$

Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence ...
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In search of a necessary condition for completeness of some metric space with application to pde

$A$ is an operator. Consider a metric space $K$ (a function $f$ is in $K$ if and only if $Af$ is in $L^2$) where the metric between two functions $f$ and $g$ is defined as $\mu (f ,g) = \int_{R^3} (...
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Closedness of first order differential operator on $L^2(\Omega)$

I am considering the when the following first order differential operator is a closed operator $$Au=b(x)\dfrac{\partial u}{\partial x_i},$$ on $L^2(\Omega)$ with the domain $D(A)=H^1(\Omega)$. Here I ...
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Nonhomogeneous Toeplitz equation

Let $T$ be the Toeplitz operator on $\ell_p$ with symbol $\alpha(\lambda)=a/2\cdot \lambda-(a+1/2)+\lambda^{-1}$, where $a$ is complex. I want to solve the following $$ Tx=y $$ for $x\in \ell_p$ ...
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Tomita Theory: Involution

Given a Hilbert space $\mathcal{H}$. Consider a von Neumann algebra: $$M\subseteq\mathcal{B}(\mathcal{H}):\quad M=M''$$ Suppose a cyclic vector: $$\Omega\in\mathcal{H}:\quad\overline{\mathcal{M}\...
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Finding the adjoint of the left translations semigroup on $L^p (\Bbb R)$

If $t \mapsto T_l (t)$ is the left translation operator by $t$ on $L^p (\Bbb R)$ given by $\Big( T_l (t) (f) \Big) (s) = f (t + s)$, find the adjoint of the left translations semigroup. Note that on $...
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functional analysis and operator theory

If the nonlinear operator N defined on R^n into itself is contraction mapping then how to show I+N is onto operator? where I is identity operator.
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Given an operator $Q$ between a Hilbert space $U$ and $L^2(ℝ^d;ℝ^d)$, is it possible to make sense of $U∋u↦(Qu)(x)$ for a fixed $x∈ℝ^d$?

Let $U$ be a Hilbert space $H:=L^2(\mathbb R^d;\mathbb R^d)$ for some $d\in\left\{2,3\right\}$ $Q$ be a Hilbert-Schmidt operator from $U$ to $H$. I want that $\tilde Q(x)$, where $$\tilde Q(x):=...
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Frechet derivative in a Hilbert space

Let $\mathcal{H}$ be a Hilbert space and $A$ a self-adjoint operator. With $(\, ,\, )$ denoting the inner product and $\psi\in \mathcal{H}$, I want to formally show that the Frechet derivative of the ...
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What's the second Fréchet derivative of a function $\mathbb R^d\to\mathbb R$

Let $u:\mathbb R^d\to\mathbb R$ be twice Fréchet differentiable. What's the second Fréchet derivative ${\rm D}^2u$ of $u$? It's clear that ${\rm D}u$ is a mapping$^1$ $\mathbb R^d\to\mathfrak L(\...
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Operator $f \mapsto u(f)$ solution of non-homogeneous Laplace equation is compact and self-adjoint

Let $u : L^2_0(D) \to L^2_0(D): = \lbrace f \in L^2 : \int_D f = 0 \rbrace $ be the linear operator which associates $f$ to $u(f)$ the solution of $$ \begin{cases} \Delta u = f & \text{in } D \\ \...
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$Tf = xf(x)$ is not compact in $L^2([0,1])$

I want to prove, in a rather elementary way, that $Tf = xf(x)$ is not compact in $L^2([0,1])$. I cannot find the appropriate bounded sequence whose image has no Cauchy sub-sequences. I have tried ...
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subset of pure states with norm condition already dense

I struggle to proof the following statement: Let $Y\subseteq P\left(B\right)$ a subset of pure states on a $C^*$-Algebra $B$ such that for every $b\in B$ there exists a $\varphi \in P \left(B\right)$ ...
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49 views

Approximate point spectrum of a normal operator

how can I show the following theorem? Let $H$ a Hilbert space and $T:H \to H$ a linear, continuos and normal operator. Then for every $\lambda \in \sigma(T)$ there exists a sequence $(x_n)_{n \in \...
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25 views

Spectral projections, additivity

Let $K$ be a positive operator on a Hilbert space $H$. $Q_1$ and $Q_2$ are projections such that $Q_1\perp Q_2$. Is $$ E^{Q_1K Q_1} (1,\infty) + E^{Q_2K Q_2} (1,\infty) =E^{Q_1K Q_1 +Q_2K Q_2} (1,\...
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28 views

Compactness of a bounded operator $A: l_1 \to l_1$

Let $l_1$ denote the space of absolutely summable sequences and $B(l_1,l_1)$ denote the space of all bounded linear operators from $l_1$ to $l_1$. I am trying to solve the following question Let $A \...
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Functions over a $C$ vector space with geometric importance. (How to find the basis?)

Searching through our suggested exercises of linear and abstract algebra for solving, I found the following exercise. The reason I am posting this, is that because we haven't went through complex ...
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Two definitions of the operator $\exp(x)$ in $L^2(\mathbb R)$

The operator $x$ acts on a dense subspace of $L^2(\mathbb R)$ and is not bounded. So if we define $\exp(x)$ via the power series $\sum_{n=0}^\infty \frac {x^n}{n!}$, convergence will not follow in the ...
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Minimality of the Spatial C*-Norm

Given C*-Algebras $A,B$ and $x \in A \otimes_* B$ show that: (a) If $(\tau \otimes_* \rho)(x)=0$ for all $\tau \in A_+^*$ and $\rho \in B_+^*$, then $x=0$. PS: $\tau \in A_+^*$ means that $\tau$ is ...
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Equivalent ways to study perturbed abstract Cauchy problem

I am considering the following abstract Cauchy problem on Banach space $X$: \begin{cases} u'(t)=Au(t)+\big(f(t)+Bu(t)\big),&t\in[0,T],\\ u(0)=x_0, \end{cases} Suppose $A$ generates a $C_0$ semi-...
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Spectrum of a polynomial operator?

Let us have $A: l^2 \to l^2, A \in B(l^2)$. $$A(\delta_n)=3 \delta_{n}+i \delta_{n+1}$$ What is the spectrum of $A$? My approach: We can write down $A$ in a better form: $$A=3I - iR$$, where $I$ ...
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Aproximating positive elements in inductive limit of C* algebras

Let $\{A_i,\Phi_{ij} \}_{i\in \mathcal{I}}$ a directed system of C* algebras and $A:=\varinjlim A_i$ its limit. I know that if $x\in A$ is self-adjoint, it can be approximated with another self-...
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Calculate the trace of $T_nL$ where $L\in L(H)$, $T\in L(H,L(H))$ and $T_n:=\langle T,e_n\rangle_H$ for some ONB $(e_n)_n$ of a Hilbert space $H$

Let$^1$ $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space over $\mathbb R$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $T\in\mathfrak L\left(H,\mathfrak L\left(H\right)\right)$ ...
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Operator groups

In $H := L^2(\mathbb{R}, \lambda)$ Hilbert-space, the following two, one-variable operator groups are given: $$(U_s f)(x):=f(x-s)$$ $$(V_s f)(x):=e^{is x} f(x)$$ $f \in H, s \in \mathbb{R}$. a, ...
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A question on Bounded Approximation property

Let $V$ be a Banach space and we say that $V$ has the $C$-BAP if there exists a net of bounded finite rank operators $T_\alpha$ in $B(V,V)$ and a constant $C$ such that $\|T_\alpha\| \leq C$ for each $...
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Operator continuity on Hilbert space

Let $A: H \to H$ be a linear operator on Hilbert space $H$, and let $\{\alpha_n\}_{n = 1}^{\infty} \subset \mathbb{R}$ converges to nonzero number. Prove that if the series $\sum_{n = 1}^{\infty} \...
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If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional?

If an operator A on a Hilbert space has compact resolvent, is Ker($\lambda-A$) finite dimensional, for any $\lambda$ in A's spectrum? P.S: What I know now is that the spectrum of A is discrete.
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Example for injective and surjective bounded and unbounded operator

I am looking for some bounded and unbounded densely defined operators on a real Hilbert space $H$, let say $A:D(A)(\subset H)\to H$, that are one-to-one but they are not onto. I am wondering whether ...
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Upper bound on the norm of the inverse of matrices with zero limit

Let $\{L(\sigma)\}_{\sigma}$ be a family of matrices indexed by the parameter $\sigma$ so that the operator norm $||.||$ of $\{L(\sigma)\}_{\sigma}$ satisfies $Ae^{-a/\sigma}\leq ||L(\sigma)|| \leq Be^...
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Functional calculus: Does $A$ commute with $e^{iA^2}$?

Let $A$ be an unbounded self-adjoint operator. Is it then true that $A$ commutes with $e^{iA^2}$? This sounds natural to me but I have no clue whether this is true in general. The problem is that we ...
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Showing a C* Algebra contains a compact operator

In my functional analysis class we are currently dealing with C* Algebras, and I just met this problem: Let $ \mathbb{H} $ be a separable Hilbert space, and suppose we have $ A \subset B(\mathbb{H}...
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Strongly continuous group and generator commute, what about square roots?

Let $A$ be a positive self-adjoint operator, then $iA$ generates a unitary strongly continuous semigroup (Stone's theorem) $T$. Then from basic semigroup theory we know that $T$ and $A$ commute, but ...
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inverse of operator

I want to calculate the inverse of the operator $T=-\frac{\partial^{2} }{\partial x^{2}}-\frac{\partial^{2}}{\partial y^{2}}-(x^{2}+y^{2})+2( x\frac{\partial }{\partial y}-y\frac{\partial }{\...
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Bochner integrability and analytic semigroup

For a general strongly elliptic second order operator of the divergence form $$A=\partial_j\big(a^{ij}\partial_{i}\big)+b^i\partial_i,$$ with smooth enough coefficients on a smooth bounded domain $\...
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Existence of Star Cyclic Vector for $M_\phi$- Necessery and sufficient condition

Let $X$ be a $\sigma$-finite measure space. $M_\phi :L_2(\mu)\rightarrow L_2(\mu)$ for $\phi \in L_\infty (\mu)$ is defined by $f \rightarrow \phi. f$. $f_0$ is called a star cyclic vector for $M_\...
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Operator Sum: Selfadjoint

Given a Hilbert space $\mathcal{H}$. Consider a closed operator: $$A:\mathcal{D}A\to\mathcal{H}:\quad A=A^{**}$$ Does it follow that: $$S:=\overline{A+A^*}:\quad S=S^*$$ (Rigorous proof?) Densely ...
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Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_n$, what's the second derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle f,...
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Suppose $f$ is a mapping between a normed space and a Hilbert space with ONB $(e_n)_{n\in\mathbb N}$, what's the derivative of $\langle f,e_n\rangle$?

Let $E$ be a normed space $(H,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a Hilbert space $f:E\to H$ be Fréchet differentiable $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ and $$f_n:=\langle f,...
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Continuity of functional calculus

Let $\mathcal{A}$ be an unital C*-Algebra. $a,b$ be normal elements in $\mathcal{A}$. $X\subset \Bbb C$ is a compact subset. $f:X\rightarrow \Bbb C$ is continuous. I need to show that for all $\...
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Neumann Poincare operator maps $L^2$ in itself

How can I show that the Neumann-Poincare operator $$ K_{\partial \Omega}[\phi](x) = \int_{\partial \Omega} \dfrac{(x-y) \cdot \nu(y)}{|x-y|^d} \phi(y) \ dy $$ maps $L^2(\partial \Omega)$ in itself (if ...
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Hilbert space …

i got the following operator. Let H be a Hilbert space, $(\lambda_n)_{n \in \mathbb{N}}$ a bounded sequence in $\mathbb{K}$ and $T^: H \to H, x \to \sum_{n\in\mathbb{N}} \lambda_n \left<x, e_n\...
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Two normal operators are similar if and only if they are unitarily similar

I need to prove that in a $C^*$-Algebra two normal operators are similar if and only if they are unitarily similar. Can anybody help, please? One side is obvious, so our concern is the other side. I ...
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Weak convergence of bounded operators

so let $X$ be a Banach space then we say that $A_n \in L(X)$ converges weakly to $A \in L(X)$ if for all $y \in L(X)^*: y(A_n) \rightarrow y(A).$ On the other hand, I just read that weak convergence ...
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35 views

Form of pure states on $M_n(\mathbb{C})$

Related question: The form of the states on an algebra of $n\times n$ matrices with complex entries I have tried to show that pure states on $M_n(\Bbb{C})$ are of the form $\phi(A)=Tr(\rho A)$, just ...
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Adjoint of differential operator

I would like to find the adjoint of the operator $T_a$ ($a\in \mathbb{C}$) on $ \mathcal{H}=L^{2}(\mathbb{R}^{2},dxdy)$ with $(u,v)=\int \int u(x,y)\overline{v(x,y)} dx dy$ $$T_a(u)=ia(y\partial_xu-x\...
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2answers
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Tridiagonal matrix inner product inequality

I want to show that there is a $c>0$ such that $$ \left<Lx,x\right>\ge c\|x\|^2, $$ for alle $x\in \ell(\mathbb{Z})$ where $$ L= \begin{pmatrix} \ddots & \ddots & & & \\ ...
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1answer
44 views

non direct sum decomposition using Fredholm operators

I am reading a paper on PDEs and at some point the author uses an argument that I cannot understand very clear, it seems to be elementary but I do not get it. Let $P$ be a Fredholm operator (i.e. ...