Questions tagged [operator-theory]

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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1answer
32 views

Associated linear system of the KP equation

The Kadomtsev-Petviashvili equation is defined as: \begin{equation} \partial_x (u_t-6uu_x-u_{xxx})=-3 \alpha^2 u_{yy} \end{equation} If we define two operators as $$ A = -4 \partial_x^3 - 6u \...
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3answers
66 views

How to prove the surjectivity of an operator

Suppose $T$ is a bounded linear operator on a banach space $X$, such that $\|I-T\| < 1$. Show that $T$ has an inverse and it is bounded. To show that an inverse exists, we have to show that $T$ ...
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0answers
56 views

Decomposition $Y=\text{im}T \oplus N$ for Fredholm operator $T$

I don't get how, from the definition of Fredholm Operator below, we get $Y=\text{im}T \oplus N$ with $N$ closed? Example 11.6(1) states Finite-Dimensional subspaces are always complemented. Example ...
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1answer
43 views

Explanation for short proof, compactness of linear operator [duplicate]

Let $H$ be a Hilbert space and $\{\phi_k\}$ an orthonormal basis of $H$. Prove that the linear operator $T:H \to H$ defined by $T(\phi_k) = \frac{1}{k+1}\phi_{k+1}$ is compact. Proof: Consider the ...
7
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1answer
153 views

Spectral theorem/ exchanging limit of series and operator

I am currently learning quantum mechanics and there is one typical scenario i encounter in my physics books: Suppose $\mathcal{H}$ is a Hilbert space and $A: \operatorname{Dom}(A)\to \mathcal{H}$ is ...
4
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0answers
36 views

Spectrum of a differential operator on $L^2(0, \infty)$

Let $A:H^n(0, \infty) \subset L^2(0, \infty) \to L^2(0, \infty)$ be the differential operator defined by $$Af:= \sum_{j=0}^na_j f^{(j)}$$ for all $f \in H^n(0, \infty),$ where $a_j \in \mathbb{C}$ and ...
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2answers
28 views

The space of bounded linear operators into a banach space is complete

This is a common theorem and is proven in many books. I am confused with a particular part of the proof. This image has been taken from Christopher Heils notes. If $A_n$ is a cauchy sequence in $B(X,...
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1answer
29 views

Confusion about equivalence of continuity and boundedness of linear operators.

I am confused with a part of the proof for the following theorem. Theorem : Let $E$ and $F$ be normed spaces. Let $T: E\rightarrow F$ be linear. Then $T$ is continuous if it is bounded. Let $T$ be ...
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0answers
29 views

Spectral measure of an eigenvector

Let $T$ be an unbounded selfadjoint operator and let $P_T$ denote it's spectral measure such that $T= \int_\mathbb{R}\lambda dP_T (\lambda)$. Suppose $\psi$ is an eigenvector of $T$ such that $T\psi=\...
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1answer
49 views

Application of Hahn Banach to show there exists $T \in BL(X,Y)$

Let $X,Y$ be Banach spaces and $U$ a closed subspace of $X$, and further $S\in BL(U,Y)$. Show in the case $Y=\ell^{\infty}$ that there exists a $T\in BL(X,Y)$ so that $T\vert_{U}=S$ and $\vert\vert ...
9
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1answer
80 views

A finite order restriction of a Fredholm operator is also a Fredholm operator.

Let $A:D(A) \subseteq H \to H$ and $B:D(B) \subseteq H \to H$ be closed linear operators on a Hilbert space $H$ such that $A$ is a finite order extension of $B$, that is, $B \subseteq A$ and $\mbox{...
2
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2answers
60 views

Consider the operator L given by $Lf(x)=\int_0^\infty e^{-tx}f(t)dt$. Show L is unbounded for 1<p<\infty

Consider the operator $L$ given by $Lf(x)=\int_0^\infty e^{-tx}f(t)dt$. View $L$ as a linear operator on the space $L^P(0,\infty)$. Show that $L$ is unbounded if $1<p<\infty$ and $p\neq 2$. Hint ...
2
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1answer
26 views

Adjoint of Toeplitz Operator and Norm-Decreasing

Let $(e_n)_{n \in \mathbb{Z}}$, where $e_n := (2\pi)^{-1/2}z^n$, be an orthonormal basis for $L^2(\mathbb{T})$ ($\mathbb{T}$ is the unit circle in $\mathbb{C}$). Let $H^2$ denote the closed subspace ...
2
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0answers
56 views

A bounded linear operator $ T: L^2 \to L^2$ with these properties (Proof check)?

A bounded linear operator $ T: L^2 \to L^2$ with these properties : 1. Commutes with translation 2. Commutes with dilation 3. Has in its kernel functions $ f $ such that support $\hat {f} \subseteq [0,...
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0answers
20 views

Confusion on when we can assume $\operatorname{dom}(T)$ is closed within itself

Say for example, we look at the sequence space $\ell^{1}$, and we define some strange norm for it, call it $\vert \vert \vert \cdot \vert \vert \vert$ (that is NOT $\vert \vert \cdot \vert \vert_{1}$) ...
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0answers
44 views

Does the space of densely defined operators on a Hilbert space form a Hilbert space?

Is it possible to define an inner product/norm on the space of densely defined (closed if necessary) operators on a Hilbert space $H$, i.e., linear maps $H\rightarrow H$? EDIT: I'm looking for a ...
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1answer
28 views

Proof Check: Continuity of Kernel Operator

Let $T_{k}: C([0,1]) \to C([0,1])$ Where $(T_{k}x)(s)=\int_{0}^{s}k(s,t)x(t)dt$ Show that $T_{k}$ is well-defined. My work: We of course simply want to show that $T_{k}x \in C([0,1])$ Let $x \in ...
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0answers
27 views

Supersymmetric Quantum Mechanics

I am reading a book on Supersymmetric Quantum Mechanics where they form the 'simplest' superalgebra with superpotentials and operators of supercharge, but I do not know what any of these words mean. ...
2
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2answers
35 views

Operator norm of translated operator

Suppose $P$ is a compact operator in $L^2([0,1])$. Assume it's contractive in the sense that $\|P\|_{\mathrm{op}}<1$. For some $f$ in $L^2([0,1])$, define the operator \begin{equation}Tg(x):= f(x) ...
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1answer
41 views

Find $\vert \vert T_{k} \vert \vert$ where $T_{k}f(s)=\int_{0}^{1}k(s,t)f(t)dt$

Let $T_{k}:L^{p}[0,1] \to L^{p}[0,1]$, where $k \in C([0,1]^{2})$ and $T_{k}f(s)=\int_{0}^{1}k(s,t)f(t)dt$ Show $\vert \vert T_{k} \vert \vert \leq \sup\limits_{s}(\int^{1}_{0} \vert k(s,t) \vert ...
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0answers
20 views

two projections with the same trace [duplicate]

Suppose $p,q\in M_n(\Bbb C)$ are two projections and they have the same trace,how to show that $dim(p(\Bbb C^n))=dim(q(\Bbb C^n))$?
2
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1answer
42 views

Extension of $*$-representation from an algebraic corner of a $*$-algebra

Let $B$ be a $*$-algebra and $A\subseteq B$ a $*$-subalgebra. Let $p\in B$ be a projection such that $$pBp=A.$$ Suppose we have a $*$-homomorphism $\phi:A\rightarrow\mathcal{B}(H)$, where $H$ is ...
3
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1answer
32 views

Showing that the $T: \operatorname{dom}(T) \to \ell^{2}$ is closed

Let $T:\operatorname{dom}(T) \to \ell^{2}$ where $T(x^{n})=(mx_{m}^{n})_{m \in \mathbb N}$ Let $\operatorname{dom}(T):=\{x^{n}\in \ell^{2}: (mx_{m}^{n})_{m \in \mathbb N} \in \ell^{²}\}$ Determine ...
1
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1answer
31 views

Weak limit of continuous operator

Let $\mathcal{A}$ be the von Neumann algebra of bounded operators on a Hilbert Space and $\mathcal{A}_{*}$ its predual. Further consider a weakly convergent sequence of continuous and bounded ...
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1answer
35 views

Non determinant -class operators satisfing $A^2+I=0$

Let $H$ be a complex Hilbert space and $A\in B(H)$ satisfies $A^2+I=0$. Does this implies that $A$ can be written in the form $A=\lambda I +T$ where $T$ is a trace classe operator and $\lambda$ ...
1
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2answers
19 views

Finding the operator norm of $\phi \in ((\ell^{2}, \|\cdot \|_{2}))^{*}$

Let $\phi(x):=\sum\limits_{n\in \mathbb N} \frac{x_{n}}{n}$ where $\phi \in ((\ell^{2}, \| \cdot \|_{2}))^{*}$ Compute: $\|\phi \|_{*}$ Let $x \in \ell^{2}$ and then $\vert \phi(x) \vert=\vert \...
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1answer
46 views

what are random matrices?

I am interested to know about random matrices and Wigner semi-circle law. Is there some easy-to-read recommendations. Possibly aimed at undergrads? non-technical stuff may help to just get a taste of ...
5
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2answers
94 views

Dixmier's lemma as a generalisation of Schur's first lemma

What mathematicians call Schur's lemma is known to physicists as Schur's second lemma: An intertwiner of two irreducible representations of a group is either zero or isomorphism. It is valid for all ...
1
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1answer
39 views

Range of self-adjoint operator

Let $T$ be a self adjoint operator in a Hilbert space $H$. Let $I$ be the identity operator on $H$ and $z\in \mathbb{R}$. Why does it hold that the range of $$T-izI$$ is $H$? Thanks in advance!
1
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2answers
58 views

Commutator of operators

I know that the commutator is defined as $[A,B]=AB-BA$, but if $A$ and $B$ are operators, how do I multiply operators? For example, if I have $$A=-4 \partial_x^3 + 6u(x,t) \partial_x + 3u_x$$ and $...
2
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1answer
43 views

Prove that there is a positive integer $m$ such that $V=\ker(T^m)\oplus\operatorname{Im}(T^m)$.

I'm studying for a qualifying exam in algebra and I've come across the following problem: Let $V$ be a finite dimensional vector space and let $T:V\rightarrow V$ be a linear operator on $V$. Let $\...
1
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1answer
24 views

Surjective (orthogonal) Projection

Let T be a compact bounded operator on a Hilbert space H. Suppose that P is a (orthogonal) projection with $P \in B(H)$ and $P(H) \subseteq T(H)$. I am trying to show that $PT: H \to P(H)$ is ...
1
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1answer
53 views

Compact operator and dual space

Let $A:Q\to Q^{*}$ be a bounded linear operator where $Q$ is a Banach space with dual $Q^{*}$. I have shown that for a sequence $\{\varphi_{n}\}$ in $Q$ with $\varphi_{n}\rightharpoonup 0$ it follows ...
1
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1answer
37 views

Proving the $x \mapsto \sum_{n}x_{n}$ is not weak-* sequentially continuous on $\ell^{1}$

Let $S: \ell^{1} \to \mathbb R$ where $x \mapsto \sum_{n}x_{n}$. Show that $S$ is not weak-* sequentially continuous (when identifying $\ell^{1}$ and $(\ell^{0})^{*}$. We are given as a hint to use ...
1
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1answer
46 views

Unital commutative Banach algebra A, A/radical(A) has no quasinilpotent elements

I am trying to show that for a unital commutative Banach algebra A, A/radical(A) has no quasinilpotent elements (where radical(A)={x\in A: x quasinilpotent}). I know that rad(A) is a closed ideal, and ...
1
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1answer
28 views

Index of tensor product of two Fredholm operators

Let $S$ and $T$ Fredholm operators on a separable complex Hilbert space $\mathcal H$ such that the tensor product $S\otimes T$ is also a Fredholm operator on $\mathcal H\otimes\mathcal H$. So what to ...
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1answer
89 views

Why trace operator is not injective?

I heard in class that the trace operator (Sobolev context!) is neither surjective or injective, and other properties that are of more importance I guess. The definitions were precisely as on Wikipedia....
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1answer
14 views

Question on convergence needed to determine upper bound for operator norm

A typical example: Let $\varphi \in (C([-1,1]),\vert\vert\cdot\vert\vert_{\infty})^{*}$ and then define $\varphi(f):=\int_{-1}^{0}f(x)dx-\int_{0}^{1}f(x)dx$ It is clear that $\vert \vert \varphi \...
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0answers
16 views

Operator theory proof for sub-multiplicativity property of distance to stationarity

For a Markov chain on discrete space $\mathcal{X}$ with transition kernel $P$ and stationary distribution $\pi$, we can define the following distance to stationarity: $$\bar{d}(t) = \frac{1}{2}\max_{(...
1
vote
1answer
27 views

Is it necessary to use Arzela Ascoli to show there is convergent subsequence of $(Tf_{n})_{n}$

Let $(f_{n})_{n}\subseteq C([0,1])$ be an arbitrary sequence such that $\vert \vert f_{n} \vert \vert_{\infty}\leq 1$ for all $n \in \mathbb N$ and $k \in C([0,1]^{2})$. Define the bounded linear ...
1
vote
1answer
45 views

How to interpret composition of Integral operators

Let $Tf(y):=\int_A k_T(x,y)f(x)dx$ and similarly for $S$ with kernel $k_S$. How do I interpret $STf(y)$? The book I'm reading states that $k_{ST}(x,z):=\int k_S(y,z)k_T(x,y)dy$. Why is that?
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2answers
44 views

How to show that $\vert \vert T \vert \vert = \sqrt{c}$ where $c:=\sum\limits_{j=1}^{\infty}\sum\limits_{k=1}^{\infty}\vert t_{jk}\vert^{2}$

Define $c:=\sum\limits_{j=1}^{\infty}\sum\limits_{k=1}^{\infty}\vert t_{jk}\vert^{2} <\infty$ and $T:\ell^{2} \to \ell^{2}$ where $(Tx)_{j}=\sum\limits_{k=1}^{\infty}t_{jk}x_{k}$ for all $j \in \...
4
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0answers
27 views

$D((A+B)^*)= D(A^*)$ if $B$ is $A$-bounded with $A$-bound $0$

Let $A:D(A) \subseteq H \to H$ and $B:D(B) \subseteq H \to H$ be linear operators on a Hilbert space $H$ such that $A$ is a closed densely defined operator and $B$ is relatively bounded with respect ...
1
vote
1answer
36 views

Functional calculus for unbounded positve operator

In Conway's book, the author has said the following (Theorem 4.10). Let $X$ be a self-adjoint positive unbounded operator. Denote $F(X)$ by the functional calculus for any Borel measurable function $F:...
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0answers
17 views

Set of tensor products

Let $A,B$ be operators that come with an argument from $L^2(\mathbb{R})$. Then we consider the set $$ \mathcal{A} = \{ e^{iA(f)}\otimes e^{iB(g)}, \ f,g\in L^2(\mathbb{R})\}$$ and in particular the ...
1
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0answers
28 views

Orthogonal function expansions and eigenvalues

Consider expanding some function $f: [-1,1] \rightarrow \mathbb{R}$ in terms of say Legendre functions: $$ f(x) = \sum_{n=0}^{N} a_n P_n(x) $$ where the truncation limit $N$ is some large number ...
0
votes
1answer
77 views

Checking injectivity of linear operator under assumptions on the adjoint

Motivation (skip if you want). I am reading a classic paper by Donaldson about orientation of Yang-Mills moduli space https://projecteuclid.org/euclid.jdg/1214441485 I am having an hard time in ...
1
vote
1answer
34 views

Surjectivity of Compact Operators on an Infinite Dimensional Vector Space

A pretty standard result says that if $T$ is a compact operator acting on a Banach space, then $T$ cannot be surjective. The proofs I've seen use the open mapping theorem and the fact that a ...
6
votes
1answer
156 views

Different definitions of a relatively compact operator

Let $T,K$ be unbounded operators on a Hilbert space $H$. I've seen the following definition of a relatively compact operator: (i) The operator $K$ is called relatively compact with respect to $T$, ...
2
votes
1answer
41 views

Determining $\| \delta_{n} \|_{*}=\frac{n}{2}$

Let $X:=C([-1,1])$ and equip it with $\| \cdot \|_{1}$. Further, let $\delta_{n}: X \to \mathbb C$ be a linear functional, such that for $n \in \mathbb N$, $\delta_{n}(f)=\frac{n}{2}\int_{-\frac{1}{n}}...