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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Gateaux Derivative.

Let $X$ be a Banach algebra. For $f\in X$, an operator $F_t$ is defined as \begin{equation} F_t(f)=\begin{cases} \Big\{\frac{f^t}{t(t-1)}, & t\neq 0,1; \\ \\ ...
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84 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
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1answer
46 views

Commutating operators and tensor products

I have this lecture slides about commutators and tensor products, but there is one part that I don't understand: The operators and are commuting operators on the tensor product and their sum has ...
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36 views

Exercise about linear operator

For $X$ Banach, I have to show that if $T\in\mathfrak{L}(X)$ and $||T||_{\mathfrak{L}(X)}<1$ then exists $(I-T)^{-1}$ and $$ (I-T)^{-1}=\sum_{n=0}^\infty T^n. $$ For the existence of $(I-T)^{-1}$ ...
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47 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
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bounded linear functional on $\ell^{1}$, and its relation to $\ell^{\infty}$

Prove that a bounded linear functional $F$ on $\ell^1$ has representation $F(x)=\sum_{n=1}^{\infty}(c_{n}x_{n})$ where $c_{n} \in \ell^{\infty}$, and that $\|F\|_{*} = \|c_{n}\|_{\infty}$.
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1answer
38 views

Linear Operator: Boundedness

I'm stuck at: $\sup_{\overline{B_1}}\lVert T x\rVert\leq\sup_{B_1}\lVert T x\rVert$? For sure it holds: $\sup_{B_1}\lVert T x\rVert=\sup_{B_1\setminus\{0\}}\lVert x\rVert\lVert T \frac{x}{\lVert ...
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1answer
30 views

Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ...
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190 views

Eigenvalues of self-adjoint eigenvalue problem

I am stack with the following problem: Consider the following eigenvalue problem $$ u \in H_B(0,1), \; \langle Lu, Lv\rangle = \lambda (\alpha \langle u, v\rangle + \langle u', v'\rangle) \; \forall ...
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$T: H^{-\infty}(R^n) \to H^\infty(R^n)$ continuous iff $T: H^{-r}(R^n) \to H^s(R^n)$ bounded for all $r,s>0$?

Denote by $H^s(\mathbb{R}^n)$ the Sobolev space on $\mathbb{R}^n$ of order $s \in \mathbb{R}$ and recall that we have $H^s(\mathbb{R}^n)^\ast \cong H^{-s}(\mathbb{R}^n)$ for the dual space of ...
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83 views

How to find adjoint operator?

Let $(X,\langle\cdot,\cdot\rangle)$ be a Hilbert Space over $K$ with orthonormal basis $(x_n)$, and let $(\lambda_n)\in K$ be a bounded sequence. The mapping $T:X\to X$ is defined by ...
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157 views

True/False: Self-adjoint compact operator

Let $H$ be a hilbert space and $T$ a compact self-adjoint operator on it. T is also injective on a dense subspace $U \subset H$ and we also have that $T(H) \subset U$. Now I am asked whether it is ...
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1answer
34 views

Closed range assumption in definition of Fredholm operators

There are two definitions of Fredholm operators (on a Hilbert space) that are commonly used. The first is that $\dim\ker T<\infty$ and $\dim\,\mathrm{coker} T<\infty$. An argument using the open ...
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51 views

Why is the image of a compact operator separable?

Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator. Question: How does it follow that the image of $S$ is separable? Thanks for the help.
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80 views

Compact kernel operator on $L^p$ space

Let $\displaystyle U_1 \subset \mathbb R^{n_1}$ and $\displaystyle U_2 \subset \mathbb R^{n_2} $ measurable sets, $\displaystyle 1 < p,q < \infty $ and consider the measurable function ...
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1answer
55 views

The trace class operators are the dual of the compact operators

I know that the map from the trace class operators $L_1(H)$ to the dual of the compact operators $K'(H)$ given by $A \mapsto tr( \cdot A)$ is an isometric isomorphism. Linearity is obvious by the ...
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Is that operator positive-definite?

Let's consider the integral operator $\phi(x) = \int\limits^1_0\psi(y)\ln\Bigl(\Bigl|\frac{\sqrt{1-x^2}+\sqrt{1-y^2}}{\sqrt{1-x^2}-\sqrt{1-y^2}}\Bigr|\Bigr)\,dy$. How to check is this operator ...
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32 views

Decomposition of Partial Isometry

I'm reading a paper and I don't understand how the operator is being decomposed. I've tried reading about different types of decomposition but nothing I read seems relevant: (Let $\mathscr{H}$ be a ...
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30 views

The product of two projections is 0

I'm reading a paper and the paper seems to think the following is obvious: Let $S$ be a semigroup of partial isometries. Let $R = \{ E \in P(S) \cup Q(S) : E$ is minimal in $P(S) \cup Q(S)$ and for ...
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54 views

Properties of an additive mappings which preserves projections

Let $A$ and $B$ be two $C^{*}$-algebras and $\Phi:A\longrightarrow B$ be an additive map which satisfies $\Phi(0)=0$, $\Phi(I)=I$ and $\Phi$ preserves projections, (i.e, $\Phi(P)=Q$ where $Q$ is also ...
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203 views

Inequalities on kernels of compact operators

Suppose we have a $\sigma$-finite positive measure $\mu(v)$ on $\Bbb R^d$ and we have two positive kernels on $\Bbb R^d\times \Bbb R^d$ $k_1(v,u)>0$, $k_2(v,u)>0$. We define integral operators ...
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112 views

Linear operators with no adjoint

Here is a standard theorem about bounded operators: Let $H$ be a Hilbert space. For any bounded linear operator $A:H\to H$ there is a unique bounded operator $A^*$ s.t $\langle Au,v\rangle=\langle ...
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Proving that two operators are equal

So I'm trying to prove that there is an equivalence between $\langle \psi\mid T\varphi\rangle=\langle\psi \mid S\varphi\rangle$ and $\langle\varphi \mid T\varphi\rangle=\langle\varphi \mid ...
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Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
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2answers
54 views

Distance between unilateral shift and compact operator

We have $S\in\mathbb{B}(\mathcal{H})$ (where $\mathbb{B}(\mathcal{H})$ is algebra of bounded linear operators in Hilbert space) and $S$ is unilateral shift. Compute ...
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1answer
97 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
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norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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1answer
57 views

A question on operator theory

Let $T$ be a quasinilpotent operator acting on a separable Hilbert space $H$. Fix a vector $x$ in $H$ such that $[T^n x]=H$ (the closed span of the orbit is $H$), and a hyperplane $Z\subset H$. Can we ...
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20 views

An operator with infinite deficiency index

I'm looking for a simple example of an operator with infinite deficiency index .
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51 views

The anti-symmetrization and simetrization operators are mutually orthogonal

For each vector $x=(x_1,\dots,x_n)$ of an $n$-dimensional vector space $V$, and for each permutation $s$ of the symmetric group on the $n$-element set $S_n$, put $s(x)=(x_{s(1)},\dots,x_{s(n)})$. Then ...
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28 views

Conditions for an Operator to Map Onto

Let operator $A[f(x)]=g(x)f(x)$ such that $A:C[a,b] \rightarrow C[a,b]$. I'm trying to think of the necessary and sufficient conditions needed on $g(x)$ such that the map is onto. Obviously it needs ...
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1answer
42 views

Multiplicative operator from L1 to L1 is given by an L_inf function

Problem: Let $\phi :X\rightarrow \mathbb{C}$ be a measurable function with respect to a measure space $(X,\mu)$. Suppose that $\phi f\in L^1(X,\mu)$ whenever $f\in L^1(X,\mu)$ and define $M_\phi ...
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113 views

Positive unbounded operators

Let $T$ be an operator in $H$. We say self adjoint $T$ is positive iff $(\forall x\in H)\langle Tx,x\rangle \geq 0 $. As in the case of bounded operators, it is true that a self-adjoint operator $T$ ...
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1answer
58 views

How to prove that the operator $(\lambda I-A)^{-1}$ exists?

Let $A:H^1(\mathbb{R})\to L^2(\mathbb{R})$ be the operator given by $Aw=w_x$, where $w_x$ denotes the weak derivative of $w$. I need help to prove that $(\lambda I-A)^{-1}$ exists and is bounded for ...
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75 views

Integral Operator Theory on $L^2[0,1]$

Let K be the integral operator on l^2[0,1] defined by itex(t) = \int_0^t (t-s)f(s)\,ds[/itex] where 0\leq t\leq 1 Show that ||K|| <1 and that tex(t)= \int_0^t ...
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57 views

Self-adjoint Hilbert Space operators

Let $H$ be a Hilbert space and $T$ is a self adjoint continuous operator in $\mathcal B(H)$. Show that $\|T^{2^{k}}\| = \|T\|^{2^{k}}$. Does this equality hold for all operators? Now it is clear that ...
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$U$ linear and bounded, is an isomorphism $\iff$ $U$ is invertible and $U^{-1}=U^*$

"Let $H$ and $G$ be Hilbert spaces and let $U:H \rightarrow G$ be a bounded operator. Prove that $U$ is an isomorphism $\iff$ $U$ is invertible and $U^{-1}=U^*$." I have denoted $U^*$ to be the ...
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3answers
30 views

Is probability function for mutually exclusive events a linear operator?

If the definitive classification criteria for a linear operator are given by: L(f+g) = L(f) + L(g) [for any/every pair of functions, f & g] L(tf) = t*L(f) [for any ...
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1answer
26 views

Showing that the square of a partial isometry is not zero

I'm reading a paper and the paper seems to imply the following is obvious: Let $S$ be a semigroup of partial isometries and suppose that $R$ is a minimal projection in the set $P(S) \cup Q(S)$ where ...
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2answers
408 views

Expectation Operator on a Matrix

Kind of embarrassing, but I'm completely blanking on what applying the expectation operator to a matrix means, and I can't find a simple explanation anywhere, or an example of how to carry out the ...
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1answer
34 views

adjoint map and dual map of complex inner product space

I know (a). but I can't solve (b) and (c). Can you help me please?
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39 views

Comparison of Symmetric Operators

The Problem: There is a unitary space $(V,<.,.>)$, $D \subseteq V $ a subspace and $ A,B : V \supseteq D\to V $ are two symmetric linear operators. Show that if: $<Ax , x> $$=$ $<Bx ...
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Find the adjoint operator of $T_p$

Let $V=\mathscr{M}_n(C)$ with an inner product $\langle A,B\rangle=\mathrm{Tr}\,(AB^{*})$, $P$ be a fixed invertible matrix in $V$, and $T_P$ be the linear operator on $V$ defined by ...
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1answer
66 views

Functions of unbounded operators: do they commute or not?

Given two unbounded commuting self-adjoint operators $A$ and $B$. Then all bounded Borel functions of $A$ and $B$ commute (in the sense that all the projections in their associated projection-valued ...
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16 views

Characterization of central (modulo the radical) elements of a Banach algebra

Let $A$ be a Banach algebra and $Z(A)=\{a \in A:ax-xa \in $ Rad $ A \ \forall x \in A\}$ be the centre (modulo Rad$A$). Then TFAE: 1) $a \in Z(A)$ 2) $\exists M >0$ such that $\rho(a+x)\leq ...
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97 views

Is the product rule true in a Banach algebra?

Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators $L:X\to X$, where the norm is given by $$\|L\|_\mathcal{L}=\sup\{\|L(x)\|_X;\;\|x\|_X=1\}$$ and the ...
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1answer
38 views

Isometry from closed operator

I have a following problem Let $H$ be a Hilbert space. We have a closed densely defined operator $A \colon D \subset H \rightarrow H$, we know that $\|Ax\| = \|x\|$ for all $x \in D$, can we extend ...
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36 views

Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all ...
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61 views

Question about normal operators

I have a question about definitions and theorems because I am a little bit confused. By definition we say that a (possibly unbounded) operator $T$ on a Hilbert space $H$ is normal if $D(T)$ is dense ...
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149 views

$\exp(A+B)$ and Baker-Campbell-Hausdorff

A few years ago, I did research in quantum mechanics, specifically dealing with generalized displacement operators. In such musings, BCH lights (or gets in, depending on your viewpoint) the way. A ...