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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7
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1answer
175 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 ...
1
vote
1answer
44 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 ...
3
votes
2answers
81 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.
3
votes
1answer
94 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 ...
2
votes
1answer
99 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 ...
0
votes
0answers
23 views

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 ...
1
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0answers
36 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 ...
1
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0answers
32 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 ...
0
votes
0answers
56 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 ...
10
votes
1answer
229 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 ...
2
votes
2answers
158 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 ...
0
votes
0answers
83 views

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 ...
5
votes
1answer
109 views

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 ...
3
votes
2answers
61 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 ...
0
votes
1answer
124 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 ...
4
votes
0answers
130 views

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 ...
2
votes
1answer
60 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 ...
2
votes
1answer
21 views

An operator with infinite deficiency index

I'm looking for a simple example of an operator with infinite deficiency index .
0
votes
1answer
70 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 ...
0
votes
1answer
30 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 ...
2
votes
1answer
49 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 ...
0
votes
1answer
193 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 $T$ ...
2
votes
1answer
59 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 ...
1
vote
1answer
84 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 ...
1
vote
2answers
75 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 ...
0
votes
1answer
62 views

$U$ 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 ...
0
votes
3answers
35 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 ...
1
vote
1answer
27 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 ...
1
vote
2answers
769 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 ...
1
vote
1answer
37 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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1answer
40 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 ...
1
vote
2answers
44 views

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 ...
0
votes
1answer
80 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 ...
0
votes
1answer
107 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 ...
0
votes
1answer
42 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 ...
1
vote
0answers
39 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 ...
1
vote
1answer
62 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 ...
6
votes
2answers
217 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 ...
3
votes
1answer
141 views

The proof that every bounded linear operator generates an unique uniformly continuous semigroup.

Let $X$ be a Banach space and $A: X \to X$ a bounded linear operator. So, $A$ is the infinitesimal generator of an uniformly continuous semigroup $\{T(t)\}_{t\geq 0}$ on $X$. The proof, as presented ...
1
vote
1answer
85 views

Preannihilator of the image of an adjoint of a bounded operator

Let $E,F$ be normed spaces and $F\colon E\rightarrow F$ be a linear bounded operator. Denote by $$A'\colon F'\rightarrow E'$$ the adjoint of the operator between the topological duals of the normed ...
0
votes
1answer
33 views

Range of the sum of two operators

Let $X$ be a Banach space, $A$ is closed densely defined operator on $X$ and $B$ is a bounded operator on $X$. Let $C$ the operator defined by $C=A+B$. I would ask about, $\mathcal{R}(C)$, the ...
1
vote
0answers
50 views

properties of integral operator $x^{-1}\int_0^xf(x,y)v(y)dy $

here we have two cases to study $(1)$ let us fix any $f \in C^{1}[ [0,1] \times [0,1]]$ ($k \neq 0$). Set $$[T(v)](x) := x^{-1}\int_0^xf(x,y)v(y)dy $$ for any $x \neq 0$ otherwise $[T(v)](0) := ...
1
vote
1answer
50 views

Sums and more on Hilbert spaces

We know that the sum of two bounded Operators is bounded and therefore also closed. But there is a fact which is more deep but i don't see the proof. Th result js the following: The sum of a closed ...
1
vote
1answer
70 views

Definition of a Bounded Operator and Some Intuition on the Definition of the Norm

I am confused about the definition of a bounded operator (which is probably a consequence of my unsatisfactory understanding of bounedeness and local boundedness). The definition is ...
0
votes
0answers
22 views

operator's domain

Let $X$ be a Banach space, $A$ is closed densely defined operator on $X$ and $B$ is a bounded operator on $X$. If we denote by $\mathcal{D}(A)$ the domain of $A$. I ask about the relation (i.e. ...
1
vote
1answer
329 views

Spectrum of operator

Like my previous question, I'm considering the same space and operator: Hilbertspace adjoint But this time I am trying to determine the spectrum of $T$. I feel like I'm messing up my definitions a ...
2
votes
2answers
86 views

restriction a non compact operator to compact operator

If $T\in\mathcal{B}(X,Y)$ is not compact can the restriction of $T$ to an infinite dimensional subspace of $X$ be compact?
2
votes
0answers
44 views

Domain of operator

Let $X$ be a Banach space, $A$ is closed densely defined operator on $X$ and $B$ is a bounded operator on $X$. If we denote by $\mathcal{D}(A)$ the domain of $A$. I ask about the relation between ...
1
vote
0answers
122 views

Partial differential equations and semigroups: explanation of an example.

The semigroup theory (as presented in Pazy's book) give us theorems that ensures existence of solutions for the abstract cauchy problem $$\left\{\begin{align*} ...
1
vote
1answer
170 views

Hilbertspace adjoint

Im doing the following excercise: Ok, so let $(e_n)$ be a orthonormal basis of $l^2$, and fix arbitrary complex numbers $(\lambda_n)$ and define $T:l^2\to l^2 $ as $$T(\sum x_ne_n)=\sum ...