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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Double sum - Miklos Schweitzer 2010

There is a question in the Miklos Schweitzer contest last year that keeps bugging me. Here it is: Is there any sequence $(a_n)$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 ...
22
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2answers
533 views

Operators with finite spectrum

Suppose that $T$ is a bounded operator with finite spectrum. What happens with the spectrum of $T+F$, where $F$ has finite rank? Is it possible that $\sigma(T+F)$ has non-empty interior? Is it always ...
20
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0answers
685 views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ ...
18
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2answers
750 views

Compact sets as point spectrum of a bounded operator

It is well known that if $K$ is any compact set in $\mathbb{C}$, then there exist a bounded linear operator $T:l_2\to l_2$ such that $\sigma(T)=K$. My questions are: Q1) Does there exist $T$, a ...
17
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1answer
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+150

Why do zeta regularization and path integrals agree on functional determinants?

When looking up the functional determinant on Wikipedia, a reader is treated to two possible definitions of the functional determinant, and their agreement is trivial in finite dimensions. The first ...
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
14
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2answers
611 views

The spectrum of an unbounded operator

It's well known that the spectrum of a bounded operator on a Banach space is a closed bounded set (and non-empty)on the complex plane. And it's also not hard to find unbounded operators which their ...
14
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1answer
197 views

$A$ and $B$ commute on a dense set but $e^{iA}$ and $e^{iB}$ do not

Let $A$ and $B$ be unbounded, symmetric operators on a Hilbert space $H$ with a common domain $D$. If $AB = BA$ on $D$, is it necessarily that case that $e^{iA}$ and $e^{iB}$ also commute? If $A$ and ...
13
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846 views

Compactness of a bounded operator $T\colon c_0 \to \ell^1$

Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact. I know how to prove this in case $\ell^r \to ...
13
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1answer
329 views

How does $\sigma(T)$ change with respect to $T$?

Consider $\sigma$ as a mapping which maps $T\in\mathcal{L}(X)$ to $\sigma(T)$, the spectrum of $T$, a compact set in the complex plane. I wonder whether there is some result concerning how ...
13
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5answers
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Operator vs function

Could someone please explain the MATHEMATICAL difference between an operator and a function? I am not talking about these in terms of coding but rather the mathematical difference. Is operator also a ...
13
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1answer
212 views

The operator $(Ax)(t)=\int_0 ^1 k(s,t) x(s)ds$

Let $X=C([0,1],\mathbb{R})$ (equipped with the supremum norm). Let $A$ be the operator defined for each $x\in X$ by $$(Ax)(t)=\int_0 ^1 k(s,t) x(s)ds,$$ where $k:[0,1]\times [0,1]\to \mathbb{R} $ is ...
12
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1answer
1k views

How to prove that an operator is compact?

Consider $T\colon\ell^2\to\ell^2$ an operator such that $Te_k=\lambda_k e_k$ with $\lambda_k\to 0$ as $k \to \infty$ how to prove that it is compact?
12
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1answer
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Operator norm and tensor norms

I have a linear operator $A\in\mathcal{L}(X,Y)$ where $X$ and $Y$ are some Banach spaces (or Hilbert spaces would also do, if that simplifies the answer.). The operator norm of $A$ is given by $$ ...
12
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2answers
264 views

Shift Operator has no “square root”?

Consider the left shift operator $T : \ell^1(\mathbb N) \to \ell^1(\mathbb N) $ by $$T(x_1,x_2..... )=(x_2, x_3 ........),$$ and also the right shift operator $S : \ell^1(\mathbb N) \to \ell^1(\mathbb ...
12
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1answer
194 views

A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the ...
12
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1answer
273 views

Singular-value inequalities

This is my question: Is the following statement true ? Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators. For every $n\in\mathbb{N}$ the following inequality holds: ...
12
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1answer
897 views

Spectrum of a linear operator

Let $\ell^2 =\ell^2(\mathbb{Z})$. Choose $\theta \in ]0,1[$ and set: $$Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{z}}$$ for each $x=(x_n)_{n\in \mathbb{Z}}\in \ell^2$ (thus $T$ is a convex ...
11
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1answer
407 views

Renorming $\mathcal{B}(\mathcal{H})$?

Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
11
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1answer
379 views

Quantization of angular momentum: is Dirac's proof wrong?

I'm trying to understand the physicist's proof of the theorem on the spectral structure of angular momentum operators (I'm being told that this proof is due to Dirac). I will refer to Ballentine's ...
11
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1answer
684 views

Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
11
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0answers
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Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 ...
10
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2answers
901 views

Gelfand-Naimark Theorem

The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that ...
10
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1answer
631 views

Importance of Toeplitz operators?

I am reading Arveson's A Short Course on Spectral Theory, in which the author states that Toeplitz operators are very important without giving references on their applications. After some searching, I ...
10
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5answers
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Integral representation for $\log$ of operator

How can one prove that $$ (\log\det\cal A=) \operatorname{Tr} \log \cal{A} = \int_{\epsilon}^\infty \frac{\mathrm{d}s}{s} \operatorname{Tr} e^{-s \mathcal{A}},$$ for a sufficiently well-behaved ...
10
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3answers
263 views

Why is such an operator continuous?

These two questions were in one question of a list of exercises. Let $E$ be a Banach space and $T : E \longrightarrow E^*$ be linear. If $\langle T(x),x \rangle \geq 0$ holds for all $x \in E$, ...
10
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1answer
201 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 ...
10
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1answer
534 views

Commuting operators and polar decomposition

Suppose that $V$ is an isometry and $X$ an arbitrary operator on a Hilbert space $H$. Let $X=U|X|$ be the polar decomposition for $X$. If $VX=XV$, can I conclude that $VU=UV$?
10
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1answer
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Every Hilbert space operator is a combination of projections

I am reading a paper on Hilbert space operators, in which the authors used a surprising result Every $X\in\mathcal{B}(\mathcal{H})$ is a finite linear combination of orthogonal projections. The ...
10
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Extension and trace operators for Sobolev spaces

Given that $\Omega \subset \mathbb{R}^{n}$ is an open, convex, Lipschitz bounded set. Let $O \subset \Omega$ be open bounded set then consider. $$u_{m} \rightharpoonup^{*} u \text{ in } ...
10
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2answers
226 views

Isometry on a dense sub-space of a Banach space?

Let $X$ be a Banach space and let $D$ be a dense sub-space of $X$. I don't know if the following fact is true: Fact: For every (linear) isometry $T\in\operatorname{Iso}(X)$ and for every ...
10
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1answer
290 views

In a C*-algebra, put $a^*a \sim aa^*$. Transitivity fails?

Idle curiosity drove me to wonder about the following question. Let $A$ be a C*-algebra. Define a binary relation $\sim$ on the cone $A^{\geq 0}$ of positive elements by putting $x \sim y$ whenever ...
10
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1answer
341 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
9
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2answers
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Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
9
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1answer
337 views

How to prove Halmos’s Inequality

How to prove Halmos’s Inequality? If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$ I found it from ...
9
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1answer
204 views

Criteria of compactness of an operator

Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$ we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$. Is it true that $K$ is compact? ...
9
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1answer
414 views

On Pitt's theorem

The famous Pitt's theorem asserts that if $p>q$ then each bounded operator $T\colon \ell^p\to\ell^q$ is compact. Since $\ell^p$ and $\ell^q$ are incomparable ($p\neq q$, $p,q\geq 1$), each operator ...
9
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1answer
75 views

How to prove $e^{\log(1+x)}= 1+x$ by series expansion?

as the title says, i want to prove $e^{\log(1+x)}= 1+x$, by substitute $\log(1+x) = \sum _{i=1} ^{\infty} \frac{(-1)^{i+1}x^i}{i}$ and $e^x=\sum _{i=0} ^{\infty} \frac{x^i}{i!} $. Can some one help ...
9
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1answer
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Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
9
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1answer
247 views

Problem 21 - Trotter theorem , Reed and Simon

This problem if from Methods of modern mathematical physics I :Functional Analysis, by Reed and Simon: Problem 21: Let $\{A_n\}$ be a sequence of selfadjoint operators on a Hilbert space $H$, and let ...
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209 views

Is there an algebra of summable series?

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : \{ Series \} \to \{ numbers \} $ be a regular, linear divergent series operator, which is either one of ...
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Are all algebraic commutative operations always associative? [duplicate]

I know that there are many algebraic associative operations which are commutative and which are not commutative. for example multiplications of matrices as associative operation is not commutative. I ...
8
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2answers
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Question about Angle-Preserving Operators

This an exercise out of Spivak's "Calculus on Manifolds". Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this. Given ...
8
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1answer
294 views

Selfadjoint compact operator with finite trace

I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$. Can we conclude that $T$ is trace ...
8
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2answers
230 views

Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...
8
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1answer
289 views

A convergence problem in Banach spaces related to ergodic theory

Suppose $X$ is a Banach space, $T\in B(X)$, satisfied the following condition. $\sup \Big\lVert\frac{1}{n}\sum \limits_{i=0}^{n-1}T^{i}\Big\rVert<\infty$ $\frac{1}{n}\lVert ...
8
votes
4answers
327 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty ...
8
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1answer
524 views

What is the use of Spectral Theorem?

Obviously the version for compact and self-adjoint linear operators on Hilbert Spaces is very useful since it decomposes the operators into orthogonal projections. However, the following more general ...
8
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1answer
265 views

Does there exist a self-adjoint operator whose spectrum consists wholly of prime numbers?

The zeros of the canonical Riemann zeta function have been compared to the prime numbers, and they have a number of special, definite connections. The infamous zeros have also been conjectured to be ...
7
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3answers
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Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?

Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain $$f(x+a) = ...