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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Reducing Spaces: Evolution

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a projection: $$P\in\mathcal{B}(\mathcal{H}):\quad P^2=P=P^*$$ Then one has: ...
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Hamiltonian: Weak Convergence

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a sequence: ...
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41 views

Laplace transform, Bochner integral

I have a quesition about linear operators on a Banach space. Let $B$ be a real Banach space. $(T_{t})_{t>0}$ is called strongly continuous contraction semigroup on $B$ if For all $t>0$, ...
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Integral kernels of self-adjoint operators

If the integral kernel $k(x, y)$ of an operator $T : C^\infty_c(M) \to \mathcal{D}'(M)$ is symmetric ($M$ is a compact manifold), then the operator $T$ is symmetric. Is the converse true? That is, ...
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27 views

Operator norm of symmetric Matrix in Hilbert Space with Hermitian Inner Product

Assume we have a postive definite real matrix $P$ and we define an inner product on a finite dimensional hilbert space $\langle x, y \rangle = x^\top P y$ and clearly the induced norm is $|| x || = ...
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Bounded linear functionals over smooth maps of a compact interval

I have two questions regarding the topological dual of the space $E = \mathcal{C}^\infty([0; 1])$ of infinitely continuously differentiable functions over the closed interval $[0; 1]$ equipped with ...
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34 views

Is there a pseudocontractive mapping that is not strictly pseudocontractive?

Given a Hilbert space $H$, a mapping $T:H\rightarrow H$ is said to be pseudocontractive if $$\|Tx-Ty\|^2\leq \|x-y\|^2+\|(x-Tx)-(y-Ty)\|^2\,\,\, \forall x,y\in H,$$ and it is strictly ...
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44 views

Operator - Exponential form

It is well known that for every unitary operator $\hat U$ an exponential of the form $$ \hat U = e^{i\hat H} $$ exists ($\hat H$ is hermitian). But I can only prove it the other way round: $$ ...
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35 views

Weak operator limit of projections in $B(H)$

Let $H$ be infinite dimensional and $\cal P$ be the set of all projections in $B(H)$. Show that $\cal P$ is weak operator dense in $(B(H))^+_{\|.\|\leq 1}$, the set of positive operators in the unit ...
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Quartic operator definition

What is a quartic operator? I googled it but found only some articles which use that term whitout giving a definition (I found that term while studying 2D Ising model, and the use of some ...
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47 views

Does a Plancherel Style Theorem for the Hardy Space $\mathbb{H}^2$ on the Unit Circle Exist?

I am working on a problem regarding Toeplitz operators, and it involves trying to prove $\mathbb{H}^2$ boundedness of the operator (defined in terms of its Fourier coefficients). Now normally when I ...
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What is the range of the operator $T$ I mean I want to determine $R(T)$

Given the normed space $\ell^\infty$ of all bounded sequences of (real or complex) numbers with the norm given by $$||x||:= \sup_{j\in Z^+} |\xi_j|,$$ for each $x:=(\xi_j)_{j=1}^\infty$ in ...
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21 views

Two self-adjoint operators with the same eigenvalues and eigenfunctions

How to show two self-adjoint operators (unbounded) on a Hilbert space with the same eigenvalues and eigenfunctions are the same.
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Unitary element in a C*-algebra

Suppose $\Bbb T$ is the unit circle and $M$ is the C*-algebra of all $2\times 2$ complex matrices. Consider the C*-algebra $A: = C(\Bbb T, M)$. Let $E$ and $F$ be the projections in $A$ given by ...
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Are symmetric matrices necessarily positive-definite / positive semi-definite?

I am trying to prove this just to be clear about this but I don't have enough conditions to force this idea to be true, so I doubt it is. Are symmetric matrices always at least positive ...
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If I want to prove that $M^{\perp}$is a closed

If I want to prove that $M^{\perp}$is a closed Can I say because it is the inverse image of $0$ by continuos function ( projection operator )
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35 views

find $\lambda$ such that the integral has a solution.

I have the integral equation: $u(x) = f(x) + \lambda \int_0^{\frac{1}{2}}u(y)dy$ I have to find $\lambda$ such that the integral has a solution. How to approach such problems?
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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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Lummer-Phillips theorem for generator of strongly continuous semigroup

Definition: Let $P_{1}\in\mathbb{K}^{n\times n}$ be invertible and self-adjoint, let $P_{0}\in\mathbb{K}^{n\times n}$ be skew-adjoint, i.e., $P^{\ast}_{0}=-P_{0}$, and let $\mathcal{H}\in ...
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44 views

Show that $C(S^n)$ is the universal $C^*$-Algebra of selfadjoint, commutative $x_0,\ldots,x_n$ with $\sum x_i^2 = 1$

Let $x_0,\ldots,x_n$ be symbols with relations $x_i=x_i^*$, $x_i x_j = x_j x_i$ and $\sum_i x_i^2 = 1$. Then I want to show that the universal $C^*$-Algebra $A$ of these relations exists and that ...
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check if a linear operator is bounded

show that $Tf = f(0)$ is not a bounded linear functional on the space of continuous functions measured with the L2 norm, but it is a bounded linear functional if measured using the uniform norm. ...
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How are $C(S^1)$ and the crossed product algebra $C(\mathbb{R})\ltimes \mathbb{Z}$ Morita equivalent?

In Connes' Noncommutative geometry one construct "noncommutative quotients" by taking certain crossproduct algebra's. Given a group $G$ acting on a set $X$ through an action $\alpha$ we can form the ...
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Given a singular matrix, I am tring to find an invertible matrix… (Finite Dimensional Space)

In coordinates and in a finite-dimensional space, how would I prove that given any singular $n$x$n$ matrix $A$, any $\epsilon\gt0$ and any matrix norm $||.||$, there is an invertible $n$x$n$ matrix ...
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Hamiltonian: Derivative

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard the evolution: $$A=A^*:\quad A(t):=e^{-itH}Ae^{itH}$$ Suppose invariance: ...
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51 views

A continuous field of C* algebra, $C(\mathbb T)\rtimes\mathbb Z_2$

Given a $C^*$-algebra, $A=${$f:[0,1]\rightarrow M_2(\mathbb C)$ where $f(0),f(1) $ are diagonal } which is isomorphic to $C(\mathbb T)\rtimes\mathbb Z_2$, How can I determine its continuous field ...
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14 views

Eigenvalue spectrum of backward Fokker-Planck operator

I encountered a paper in physics, in which the author states that an operator of the following form (backward Fokker-Planck) $\Lambda = P(x)\frac{d}{dx}+Q\frac{d^2}{dx^2}$ has an eigenvalue 0 and ...
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38 views

Convergence in strong operator topology and norm topology

Let $(T_n)\subset B(H)$ be a sequence of operators such that $T_n\to 0$ in strong operator topology. Show that $\|T_nK\|\to 0$ and $\|KT_n\|\to 0$ for every compact operator $K$. Let $f,g \neq 0 ...
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Show that $ T \notin X' $ if $ X = C([0,1]) $ is equipped with the norm $ \| f \|_{L^{2}} \stackrel{\text{df}}{=} \sqrt{\int_{0}^{1} |f|^{2}} $.

Let $ T $ be an operator on $ X = C([0,1]) $ defined by $ T(f) \stackrel{\text{df}}{=} f(0) $. I want to show that $ T \notin X' $ (the dual space of $ X $) if $ X $ is equipped with the norm $$ \| f ...
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54 views

Closed unit ball of positive bounded operator space and its extreme point

Let $H$ be infinite dimensional Hilbert space. Then the closed unit ball of positive bounded operator space $B(H)^+$ is not the convex hull of the projections of $B(H)$. Please help me. Thanks.
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The spectrum of a self-adjoint operator on $\mathcal l^2$

Let $S$ be the unilateral shift operator on $\mathcal l^2$ (which shifts one place to the right) and $S^*$ its adjoint, the backward shift (which shifts one place to the left). I've been trying to ...
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1answer
18 views

$\max$ Operator on a Sum of Integrals

Define the value function $$v(k) := \max_c\left[\int_0^t{F(s,c,k)ds}+\int_t^\infty{F(s,c,k)ds}\right]$$ Is this expression equal to $$v(k) := ...
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Linear operator differentiation on a torus

I'm trying to analyze this article about area-preserving diffeomorphisms and don't quite understand a sentence. 4.1. Linear involutions. We start characterizing the linear involutions $R \! : ...
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An extension of a corollary to Fuglede's theorem

Fuglede's theorem states that if $T,N\in B(H)$ for some Hilbert space $H$ and $N$ is normal and $TN = NT$, then $TN^* = N^*T$. A corollary to this theorem is that if $M,N \in B(H)$ are normal and ...
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Hamiltonian: Commutator

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote for shorthand: ...
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Sesquilinear Forms: Hamiltonian (II)

Given a Hilbert space $\mathcal{H}$. Consider a positive form: $$s:\mathcal{D}\to\mathcal{H}:\quad s(\varphi,\varphi)\geq0$$ Introduce its form space: ...
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53 views

A nonzero compact operator on a Hilbert space has a nonzero eigenvalue

Let $T:H\to H$ be a compact operator on a Hilbert space $H$, with $T\neq0$. Prove that $\exists c\neq0$ and $x\neq0$ such that $Tx=cx$. I was trying to prove it using the fact that if $T$ is a ...
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Integral Operator Contraction

I have the following question: I've found the bound as follows $\lvert T f(x)\rvert \le C\lVert f \rVert(x-a)$ using the fact that K is bounded on the closed square so we have $max \lvert K(x,y) ...
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Invariance of semigroups

$A$ is the infinitesimal generator of the $C_{0}$-semigroup $(T(t))_{t\ge 0}$ and $V$ is a one dimensional linear subspace of $X$. I want to show that $V$ is $T(t)$-invariant $\iff$ ...
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41 views

Sesquilinear Forms: Hamiltonian (I)

Given a Hilbert space $\mathcal{H}$. Consider a dense positive form: $$s:\mathcal{D}\times\mathcal{D}\to\mathbb{C}:\quad s(\varphi,\varphi)\geq0\quad(\overline{\mathcal{D}}=\mathcal{H})$$ Construct ...
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48 views

Introducing an operator by a bilinear form

Let $I = (0, 1)$ and $H_0^2 (I)$ the closure of $C_c^\infty (I)$ in $W^{2, 2} (I)$. Consider $$a : H_0^2 (I) \times H_0^2 (I) \to \mathbb{R}$$ defined by $$a(u, v) = \int\limits_I u''(x) v''(x) \, ...
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Spectrum of operator $Af(t) = \int_0^{t^2} f(s)ds$ on $L^2[0,1]$

Consider a linear operator $A\colon L^2[0,1]\rightarrow L^2[0,1]$ that acts as follows: $$Af(t) = \int_0^{t^2} f(s)ds$$ The problem is to compute its spectrum. I know that the operator is compact ...
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Are elements of a $C^*$-Algebra strictly positive iff their spectrum is strictly positive?

Let $A$ be a $C^*$-Algebra. An element $a\in A$ is said to be positive iff $a=a^*$ and the spectrum $\sigma(a)$ is nonnegative, ie. $\sigma(a)\subset[0,\infty)$. This is equivalent to $\varphi(a)\ge ...
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Congruence Property of Monotone Operators

A map $T$ is called strictly monotone if for $x\ne y$, $\langle u-v,x-y\rangle>0$ for all $u\in T(x),v\in T(y)$. Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if ...
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weak closure of unitary group in $B(H)$

Let $H$ be a Hilbert space with dim $H=\infty$ , and $\cal{U}$ be the group of all unitaries on $H$. Show that the weak closure of $\cal{U}$ is a semigroup with respect to the multiplication. I know ...
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(soft question) why do we study complex measures and complex-valued functionals in modern analysis

Recently I am struggling with "complex" things for my "real" analysis class. We are using Folland's Real analysis, 2nd for text book. It seems that Folland is trying to use complex-valued functions ...
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Proving strong stability of semigroup

$X$ is the Hilbert space $L^{2}(0,\infty)$ and let $T(t):X\to X$ with $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$. I want to prove that the $C_{0}$-semigroup $(T(t))_{t\ge 0}$ is strongly ...
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Bounding the operator

If $X$ is the Hilbert space $L^2(0,\infty)$ equipped with the inner product $$\langle f,g\rangle :=\int_0^\infty f(\zeta)\overline{g(\zeta)}(e^{-\zeta}+1) \, d\zeta,$$ and the operator $T(t):X\to ...
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102 views

Generating a contraction semigroup on an energy space

Consider the system of partial differential equations $\displaystyle\frac{\partial Q}{\partial t}(\zeta, t)=-\frac{\partial}{\partial\zeta}\frac{\phi(\zeta, t)}{L(\zeta)}$ ...
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Schrodinger Operator with Finite Discrete Spectrum in $(-\infty, -1]$

I'm reading parts of Reed and Simon's Analysis of Operators and have come across a statement I find puzzling. They say that if $V$ is a bounded function of compact support on $\mathbb{R}^3$ then ...
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1answer
40 views

closed graph theory and unbounded operator

I am studying unbounded operators and the graphs of those operators. I found that the closure of a graph may not be the graph of any operator. Can someone provide an example of an operator and a ...