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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How to generalize this proof of the closed graph theorem

I found this tricky new proof of the closed graph theorem for a Hilbert space $H$. http://arxiv.org/pdf/1601.02600.pdf It says in the abstract, that it's possible to extend the proof to Banach space. ...
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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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1answer
39 views

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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1answer
22 views

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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1answer
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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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Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentially selfadjoint if it contains a dense subset of analytic vectors?
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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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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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1answer
29 views

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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124 views

Show that operator is continuous and selfadjoint (or not)

In this thread Show compactness/ noncompactness of an operator by approximation I came to the conclusion that the operator $$ T\colon\ell^2\to\ell^2, (a_n)_{n\in\mathbb{N}}\mapsto\left(\frac{a_n+a_{n+...
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1answer
33 views

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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43 views

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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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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1answer
57 views

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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21 views

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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1answer
45 views

$Tf = xf(x)$ 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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Question about Angle-Preserving Operators

This an exercise out of Spivak's "Calculus on Manifolds". Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this. Given $x,y\in\...
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48 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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60 views

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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1answer
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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1answer
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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112 views

Can we characterize the probability generating function as a linear operator?

For a nonnegative integer-valued random variable $X$ with $\mathbb P(X=j)=p_j$, we define the probability generating function of (the distribution of) $X$ by $$P_X(s):=\mathbb E\left[s^X\right] = \...
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1answer
31 views

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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20 views

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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1answer
24 views

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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22 views

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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52 views

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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32 views

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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1answer
26 views

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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1answer
18 views

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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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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22 views

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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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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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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1answer
22 views

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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13 views

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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1answer
47 views

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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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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1answer
57 views

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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65 views

Is the Hankel Transform a Hankel Operator

The "Hankel Transform" is the infinite weighted sum of the Bessel function. At the top of the wikipedia article http://en.wikipedia.org/wiki/Hankel_transform it says Not to be confused with the ...
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82 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
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Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal X,...
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1answer
153 views

Fractional powers of positive self-adjoint operators

Consider two positive unbounded operators $A$ and $B$ densely defined on a Hilbert space $H$ self-adjoint on a domain $\mathcal{D}(A) = \mathcal{D}(B) = H_1$. By the spectral theorem, we can define ...
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1answer
22 views

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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1answer
33 views

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,...
2
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1answer
104 views

Why is $\langle Ax, Ax \rangle = \langle A^2 x, x\rangle$?

Let $X$ be a Hilbert space and $A\in \mathcal{B}(X)$ be self-adjoint. How can I prove: $$\langle Ax, Ax \rangle = \langle A^2 x, x \rangle$$ I know it is a simple problem, but I don't know how to ...
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1answer
24 views

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,...