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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If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
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46 views

The hermitian element $h=\sum_{n=1}^\infty \frac{p_{n}}{3^{n}}$ generates $C_{0}(\Omega)$‎

‎Please help me to solve the following problem‎ : Let $\Omega$ be a locally compact Hausdorff space‎, ‎and suppose that the $C^{*}$-algebra $C_{0}(\Omega)$ is generated by a sequence of projections ...
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90 views

Exponential of an operator plus a constant term

I am reading a book on operator and matrix representation. Most of the examples are on Physics and they mention many terms like 'commute', i.e. the order of application of two operators might be ...
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99 views

Finding the spectral radius and spectrum .

I am solving the following question : If $k:[0,1]^2\to \mathbb C$ is continuous and $T_k : C[0,1] \to C[0,1]$ such that $$(T_kx)(t)=\int_0^t k(t,s)x(s) ds$$ Define $k_n: [0,1]^2\to \mathbb C$ ...
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29 views

How to show that density?

Show $$ \overline{\operatorname{span}(v_j)}=L^2([0,1]),~~~~~\overline{\operatorname{span}(u_j)}=L^2([0,1]) $$ with $$ v_j(x)=\sqrt{2}\cos((j-1/2)\pi x),~~~~~u_j(x)=\sqrt{2}\sin((j-1/2)\pi x). $$ ...
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83 views

Extension of differentiation operator to $L_2[0,1]$.

I'm studying for my functional analysis exam. We are required to know the proof of the following, but I cannot figure it out. Consider $L_2[0,1]$ with orthonormal basis $(e_n)_{n=-\infty}^\infty$ ...
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65 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
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65 views

Positive maps on $\mathcal{B}(\mathcal{H})$ to itself

Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
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113 views

Composition of positive maps

Let $\chi_A:\mathcal{B}(\mathbb{C}^n)\rightarrow\mathcal{B}(\mathbb{C}^n)$ be a completely positive (cp) map defined as $\chi_A(x)=AxA^*$, where $A\in\mathcal{B}(\mathbb{C}^n)$. Clearly any cp map ...
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249 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
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47 views

Similarity orbit of compact operators

I am considering a problem connecting the spectra of compact operators to larger class of operators. Since spectra are invariant under similarity, I wonder whether there is a good reference on ...
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93 views

When the ultrastrong closure of a *-algebra contains the double commutant

As lemma 6 on p.44 of Dixmier's book on Von Neumann algebras, he states that if $A$ is a *-algebra (i.e. possibly without identity, not necessarily closed in any topology) of operators in $B(H)$ such ...
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134 views

Fixpoint of monotone operators

Let $X$ be some set and let $F$ be the set of all functions with a domain $X$ and a range $[0,1]$. We consider $F$ to be a partially ordered set with $f\leq g$ if and only if $f(x)\leq g(x)$ for all ...
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192 views

Convergence of net sums of complex numbers, as well as operators

I have some questions concerning convergence of sums where the summands are complex number, although the real motivation of my question comes from Von Neumann algebras where sometimes the summands are ...
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233 views

Orthogonal projection and normal operators

Let $G$ be normal operator with compact resolvent such that $\ker G$ is different from $\{0\}$. Now Let $P$ be the orthogonal projection onto $\ker G$ and consider $G' = G + P$. Please, I want an ...
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69 views

Is there chance to form a frame (Riesz basis)?

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ One can show that ...
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66 views

When is the orbit of a vector a minimal sequence? When does an operator have a minimal orbit vector?

Let $X$ be a banach space. We say a sequence $(x_n)$ is minimal if for each $k$, $x_k\notin [x_n]_{n\neq k}$, where $[x,y,z,\cdots]$ is the closed linear span of the vectors. For ...
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84 views

Where to find Kelly's thesis on Weighted Shifts on Hilbert Space?

I am reading about weighted shifts on a hilbert space. So many of the books/ papers list R.L. Kelly's paper Weighted Shifts on Hilbert Space as a reference that I really want to have a look at this ...
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107 views

Partition of unity. Functional analysis

Need to find partition of unity in case of oparator $A_{f}(x)=(|x-1|+x)f(x)$. Operator $A \in L_{2}[0,2]$ Partition of Unity is set of operators $E_{\lambda}=E((- \infty,\lambda]) $, where ...
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85 views

positively homogeneous asymptotic expansion associated to the symbol of a pseudodifferential operator

I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic ...
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152 views

Neumann series in an incomplete normed algebra

Let $\mathcal{A} \equiv (A, \|\cdot\|_A)$ be a unital (associative) normed algebra over the real or complex field, and assume that $\mathcal{A}$ is not complete. Provided $\mathcal{B}_\mathcal{A}$ is ...
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109 views

How can I compare unbounded linear operators?

Let $X$, $Y$ be Hilbert spaces. Let $S, T : X \rightarrow Y$ be unbounded operator. Suppose $S$ and $T$ be bounded operators. Then we can compare by their maximum distance on the unit ball of $X$. ...
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165 views

Operator functions

Can anyone give me an example/application where a selfadjoint bounded operator on a Hilbert space are put through a function? I know how to define it but am struggling to find an application area for ...
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20 views

Derivatives of Differential operators

Let $V=H^1(\Omega)$ and $U = L^{\infty}(\Omega)^2$. Take for example, a differential operator, $L:U\times V \rightarrow V^* $ such that: \begin{equation} L(u,y) = \Delta y + \vec{u}(x)\nabla y ...
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9 views

How to show a Borel Operator Measure dilates to a Spectral Measure?

Does anyone know a simple proof of the following theorem stating that a positive Borel operator measure $P$ on $\mathbb{R}$ can be written as $V^{\star}EV$ for a Borel spectral measure $E$? ...
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24 views

Spectral Measures: Spectral Subspaces

Given a Hilbert space $\mathcal{H}$ and let the Lebesgue measure be $\lambda$. Consider a normal operator $N:\mathcal{D}\to\mathcal{H}$. Denote its associated Borel spectral measure by: ...
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14 views

Numerical range of closure of operator

Let $B$ be an unbounded densely defined and closable operator. If $\mathcal{N}(B)$ is the numerical range, what can be said about the numerical range of its closure $\overline{B}$?
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20 views

computation example of Maximal operator

how to compute the answer M(f(x))? can anyone show a detail step? thank you!!
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12 views

spectrum of a convolution operator is connected

Let $k \in L^1(\mathbb{R}; \mathbb{C})$ and define a linear operator $T$ acting on $L^2(\mathbb{R}; \mathbb{C})$ by $$ Tf(x) := [k \ast f] (x) \ldotp$$ Linearity is obvious, while the fact that $T$ is ...
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20 views

Why this operator is not fredholm?

Define $f:S^{2}\to \mathbb{R}$ by $f(x,y,z)=z$. Let $D:=D_{\nabla f}$. As I learned from the following post this operator is not counted as a fredholm operator.( I did not underestand, ...
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16 views

Show that $H=\oplus_{\alpha \in \sigma(T)}\text{ker}(\alpha I -T)$

Suppose $T$ is an operator on a Hilbert space $H$ such that $\sigma(T)=\sigma_{e}(T)$, and for each $\alpha \in \sigma(T)$, the corresponding eigenspace ker$(\alpha I-T)$ is a reducing subspace for ...
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30 views

Tensor products, help with proof

Let $X$, $Y$ and $Z$ be Banach spaces. Let the space $X\otimes_{\epsilon}Y\otimes_{\epsilon}Z^{*}$ be the injective tensor product. The injective norm is defined as follows: ...
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15 views

Notation for operator that returns square of a function?

Let $F$ denote the vector space of all real-valued continuous functions on the real line. Suppose I have an operator $T:F\to F$ such that for any input function $f \in F$, $T$ returns the square ...
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6 views

Monotone operator without symmetry?

A function $f : \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ is monotone with respect to $P = P^\top\succcurlyeq 0$ if $$ \left( f(x) - f(y) \right)^\top P (x-y) \geq 0 $$ for all $x,y$. Now suppose that ...
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10 views

Adjoint Operator and transposed linear application

I have some problems to well understand the notion of adjoint. 1) First to simplify, let's choose an operator $u:E \to F$ defined on the whole domain from E (banach space) to F (Banach space). It ...
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11 views

Limit of an element in a unital C*-algebra

Let $A$ be a unital C*-algebra. Show that an element $x$ of $A$ is self-adjoint if and only if $\lim_{t\to 0}\frac{1}{t}(||1+itx||-1)$=0. My attempt: Suppose $x=x^*$. By functional calculus of x, ...
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31 views

Confusion with Riesz

I was reading this article here and it claims that the covariance operator from a Hilbert space to itself exists by the "Riesz representation theorem". I don't seem to see the link between the Riesz ...
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21 views

Find an operator $Z$ in $H^1(0, \infty)$ with $\langle u,Zv\rangle = \int \bar{u}v dx$

I'm working with operators associated to bilinear forms. What I need to find is a continous, linear operator $T$ defined on $H^1((0, \infty))$ [note that $H^1 = W^{1,2}$ is the Sobolev space] such ...
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24 views

Derivative of a function which is treated as a variable

I have got a function $f=f(x)$. The derivative is $\partial_xf$. There are applications in which it is reasonable to treat $f$ as another variable in a larger context. In my application I now need an ...
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11 views

Homeomorphism between locally compact space $\Omega$ and maximal ideals space of $C_0(\Omega)$

the following is a proposition: If $\Omega$ is locally compact and $\Sigma$ is the maximal ideal space of $C_0(\Omega)$, then the map $x\to \delta_x$ is a homeomorphism. To prove it, the author ...
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51 views

Closed unit ball of B(H) is not compact in strong operator topology of B(H).

In operator theory we prove that closed unit ball of B(H) is compact in weak operator topology and is closed in strong operator topology. But a book of operator theory states that closed unit ball of ...
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60 views

Does the integral operator, whose kernel is the indicator of the rhombus, belong to the trace class?

In connection with this question: Does the integral operator on $L^2(\mathbb R)$, whose kernel is the indicator of the rhombus $\{|x|+|y|<1\}$, belong to the trace class?
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40 views

Notation of measures $d \mu$

I am reading the paper http://www.ams.org/mathscinet-getitem?mr=3246935 and there some notation that I have found a bit confusing on page 1503 between Lemma 4.2 and 4.3. I'll give as much context as I ...
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40 views

Every homomorphism on a C*-algebra is a *-homomorphism

The following is a proposition of Conway's Functional analysis: and also he uses below exercise to proof above proposition: But I do not know how he uses the Exercise and say $||h||=1$. Please ...
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24 views

Relation between two spectra

This seems like an easy enough computation but I'm stuck! Let $X \in B(H)$ for a Hilbert space $H$ such that $X^{2}=0$, but $X\neq 0$. With respect to the decomposition $H=\text{ker}X \oplus ...
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29 views

How to show an operator is invertible?

How to show an operator is invertible in an abstact setting? I only know that if $\|T\|< 1$ then $I-T$ is invertible. For example: Let $(X_1, \|\cdot\|_1)$ and $(X_2, \|\cdot\|_2)$ be Banach ...
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28 views

Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...
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20 views

Composition operator of a non-analytic function

I recently came across a problem that can be reformulated in simple terms involving the composition operator $$ C_g\colon X \to X, \quad f \mapsto f \circ g $$ for functions $$ f,g\colon \mathbb{C} ...
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35 views

Exercise of direct sum of operators: could someone please check my work

I tried to do this exercise and was wondering of someone could please read my work and tell me if it is correct: Let $u: X \to Y$ and $u': X' \to Y'$ be bounded linear operators between Banach ...
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15 views

Is there an expression for $\exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = $?

Does an expression for $$ \exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = ? $$ exist? For j=1 we have the usual expression for translation and scaling $$ \exp\left( t \partial_z\right) f(z) = ...