Questions tagged [operator-theory]

Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

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Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...
Emilio Pisanty's user avatar
47 votes
0 answers
2k views

Set-theoretic reformulation of the invariant subspace problem

The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
Nik Weaver's user avatar
39 votes
3 answers
13k views

Is the Invariant Subspace Problem interesting?

There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...
William DeMeo's user avatar
34 votes
1 answer
3k views

tr(ab)=tr(ba), part 2.

This is a Banach space version of Andre Henriques' question Trace Question for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
Bill Johnson's user avatar
32 votes
3 answers
3k views

Why are there so many fractional derivatives?

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator. I started with the book The Fractional Calculus ...
FusRoDah's user avatar
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31 votes
0 answers
2k views

Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?

Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
Mikhail Ostrovskii's user avatar
30 votes
3 answers
5k views

When is an integral transform trace class?

Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator $$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$ the operator $K$ is Hilbert ...
Marc Palm's user avatar
  • 11.1k
27 votes
0 answers
1k views

Unital $C^{*}$ algebras whose all elements have path connected spectrum

A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. What is an example of a non commutative ...
Ali Taghavi's user avatar
26 votes
2 answers
5k views

Understanding a simplifying assumption in proof of the invariant subspace problem

In a recent preprint On the invariant subspace problem in Hilbert spaces Per H. Enflo claims to have solved the invariant subspace problem, showing that every bounded linear operator on a separable ...
Federico's user avatar
  • 423
20 votes
3 answers
7k views

Why do inner products require conjugation?

For Hermitian matrices and operators, the most "natural" inner product is $f^H \cdot g$ or $\int f^* g\; dx$. A similar situation holds interpreting Fourier transforms as the inner product of ...
Victor Liu's user avatar
20 votes
0 answers
301 views

Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$

Remark: I cross-posted this question on MSE and added a bounty to it. Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
Calculix's user avatar
  • 321
19 votes
1 answer
2k views

Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...
William Bell's user avatar
18 votes
1 answer
551 views

Is the space of Hankel operators complemented in B(H)?

Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$. Let ...
Yemon Choi's user avatar
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18 votes
4 answers
1k views

Who first used the multiplication operator version of spectral theory

This is another history question. Hilbert phrased the spectral theorem in terms of resolutions of the identity. While this remained the form of Stone and von Neumann, they did also have the ...
Barry Simon's user avatar
17 votes
1 answer
2k views

Are "most" operators on an infinite-dimensional complex Banach space "diagonalizable"?

This is true for finite-dimensional spaces: the diagonal operators on a finite dimensional complex vector space form contain a dense open set and the nondiagonalizable operators have measure 0. To be ...
Tim Campion's user avatar
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17 votes
2 answers
2k views

The letters of the word "ART"

Edit: According to the Gelfand duality between topological spaces and commutative $C^{*}$algebras, I add some new tags. So the question is that what is the structure of $ Ext (A,A)$ where $A$ is $...
Ali Taghavi's user avatar
17 votes
2 answers
936 views

Convexity of spectral radius of Markov operators, Random walks on non-amenable groups

Let $P_1,P_2$ denote stochastic transition matrices on a countable set $I$. Consider $P_1,P_2$ as operators on $\ell^2(I)$ given by multiplication. Question Under which conditions can we show that ...
Mika's user avatar
  • 171
16 votes
2 answers
2k views

Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?

Consider the operator $\frac D{e^D-1}$ which we will call "shadow": $$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
Anixx's user avatar
  • 9,306
16 votes
1 answer
1k views

Hodge de Rham operator and orientability

Let $(M,g)$ be a Riemannian manifold. One can consider the exterior algebra bundle $\Lambda(T^*M)$. The sections of this bundle are differential forms, to be noted by $\Omega^k(M)$. One can consider ...
truebaran's user avatar
  • 9,140
15 votes
3 answers
1k views

Version of Banach-Steinhaus theorem

I am wondering about the following version of the Banach-Steinhaus theorem. Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
Sascha's user avatar
  • 506
15 votes
2 answers
791 views

An orbit of symmetric polynomials

Consider the ring of polynomials $R:=\mathbb{Z}[x_1,x_2,x_3]$. Define the operators $E, I:R\rightarrow R$ by $Ef(x_1,x_2,x_3)=f(x_1-1,x_2,x_3)$ and the identity $If=f$. Let $\mathcal{L}:R\rightarrow R$...
T. Amdeberhan's user avatar
15 votes
2 answers
634 views

Multiple of identity plus compact

Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...
user129564's user avatar
15 votes
1 answer
618 views

Approximate eigenvectors for a set of non-commuting self-adjoint operators

This problem is motivated by finding the right mathematical setting for expressing the compatibility of classical physics with quantum mechanics. Let $\mathcal H$ be a Hilbert space and $S$ a ...
David Mumford's user avatar
15 votes
0 answers
2k views

PT Symmetry and the Riemann Hypothesis

Recently there have been articles in Quanta, in Science Alert, and at phys.org among others, on possible recent progress toward the Hilbert-Polya conjecture, which implies the Riemann Hypothesis. The ...
Stopple's user avatar
  • 10.8k
14 votes
5 answers
2k views

How far is a set of vectors from being orthogonal?

Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones? Or, more formally... Suppose $...
Louis Deaett's user avatar
  • 1,513
14 votes
4 answers
4k views

How much does the absolute value of an operator behave like an absolute value?

Recall that the absolute value of a bounded operator $T$ on a Hilbert space $H$ is the unique positive operator $|T|$ such that $$\||T|x\|=\|Tx\|$$ for all $x\in H$. It can be defined using the ...
Iian Smythe's user avatar
  • 3,001
14 votes
1 answer
651 views

Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional. Question. Does it follow that $A$ is finite dimensional? ...
Jochen Glueck's user avatar
13 votes
2 answers
1k views

Homotopy groups of Fredholm operators

If $X$ is separable complex Hilbert space and $\mathcal{F}$ the topological space of Fredholm operators on $X$, then it is well-known, that $$ \pi_0(\mathcal{F}) = \mathbb{Z}\, , $$ i.e. the connected ...
Chandler's user avatar
  • 173
13 votes
5 answers
2k views

If two projections are close, then they are unitarily equivalent

Given two projections $p,q\in B(H)$, it is well-known that if $\|p-q\|<1$, then there exists a unitary $u\in B(H)$ with $q=upu^*$. The proof that immediately occurs to me uses comparison of ...
Martin Argerami's user avatar
13 votes
2 answers
886 views

Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?

Let $X$ and $Y$ be Banach spaces. The notion of the embedded spaces was introduced by D.S. Djordjevic. Say that $X$ embed in $Y$, and write $X \preceq Y$, if there exists a left invertible operator $...
Qingping Zeng's user avatar
13 votes
1 answer
572 views

Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) theme are there?

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal ...
მამუკა ჯიბლაძე's user avatar
13 votes
1 answer
1k views

Is there an algebra for divergent series summation operators?

Let $D$ denote a divergent series and let $C$ denote a convergent series. Furthermore, let $s : $ { Series } $\to$ $\mathbb{C}$ be a regular, linear divergent series operator, which is either one ...
Max Muller's user avatar
  • 4,485
13 votes
1 answer
457 views

One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8. Suppose ...
DLIN's user avatar
  • 1,905
12 votes
6 answers
7k views

Is there an analog of determinant for linear operators in infinite dimensions as that of finite dimensions?

I am trying to find out the essence of what a determinant is. Besides, in finite dimensions, determinant is the kind of numerical invariant that determines the invertibility of a linear operator, but ...
Xuxu's user avatar
  • 663
12 votes
3 answers
864 views

Bibliographic request concerning an article by Bernstein and Robinson

Concerning the article "Bernstein, Allen R.; Robinson, Abraham. Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in finding ...
Mikhail Katz's user avatar
  • 15.1k
12 votes
1 answer
876 views

Is there a proof that the $C^{*}$-algebras don't see the invariant subspace problem?

This post is an appendix of this one. Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is ...
Sebastien Palcoux's user avatar
12 votes
3 answers
1k views

Is the set of separable quantum states closed?

Let $\mathcal H,\mathcal H'$ be Hilbert spaces (not necessarily separable). A "separable state" is a trace-class operator of the form $\sum_i \rho_i\otimes\rho_i'$ where $\rho_i,\rho_i'$ are positive ...
Dominique Unruh's user avatar
12 votes
2 answers
502 views

Existence of closed operators with arbitrary dense domain of a given Banach space

Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., $D(T)=Y$?...
Tomas's user avatar
  • 799
12 votes
1 answer
1k views

Decomposition of positive definite matrices.

It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum $$ A=\sum_{j} B_j \otimes C_j $$ with $B_j$ and $C_j$ positive semidefinite matrices (of ...
Ruben A. Martinez-Avendano's user avatar
12 votes
1 answer
184 views

Spectra on different spaces

This is a method request: I am looking for techniques that allow me to investigate problems like this: Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\...
Kinzlin's user avatar
  • 295
12 votes
0 answers
149 views

Holomorphic natural bundles and operators

I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting. After a quick thought, I've gone through the standard ...
Quidam_t's user avatar
  • 121
11 votes
2 answers
2k views

Operator that commutes with projections

We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$ Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...
Sascha's user avatar
  • 506
11 votes
3 answers
3k views

Dual space of $L^2(\mathbb{R},L^1(0,1))$?

I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures) Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
Jacob Augstine's user avatar
11 votes
3 answers
433 views

Does the generator of a 1-parameter group of Banach space isometries know which elements are entire?

Let $X$ be a complex Banach space. Let $(\sigma_t)_{t \in \mathbb{R}}$ be a 1-parameter group of linear isometries of $X$ which is strongly continuous i.e. $t \mapsto \sigma_t(x)$ is continuous for ...
Michael's user avatar
  • 672
11 votes
2 answers
1k views

What's the matrix of logarithm of derivative operator ($\ln D$)? What is the role of this operator in various math fields?

Babusci and Dattoli, On the logarithm of the derivative operator, arXiv:1105.5978, gives some great results: \begin{align*} (\ln D) 1 & {}= -\ln x -\gamma \\ (\ln D) x^n & {}= x^n (\psi (n+1)-\...
Anixx's user avatar
  • 9,306
11 votes
2 answers
1k views

Concentration compactness. Can this concept be stated in a theorem?

I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk. When I approached the speaker ...
Zinkin's user avatar
  • 491
11 votes
2 answers
702 views

A neat evaluation of an infinite matrix?

Let $M_n$ be an $n\times n$ matrix defined as $$M_n =\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$ With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
T. Amdeberhan's user avatar
11 votes
2 answers
346 views

Exponential decay of voltage potential difference

Consider the following adjacency matrix of a complete graph: $$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$ with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
neverevernever's user avatar
11 votes
2 answers
544 views

Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra?

Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by $(zf)(g):...
Werner Thumann's user avatar
11 votes
1 answer
1k views

Quasi-nilpotent trace class operators as limits of nilpotents

In as yet unwritten work with T. Figiel and A. Szankowski we make an observation in a Banach space context that for Hilbert spaces reduces to: If $T$ is a quasi-nilpotent (i.e., has only $0$ in its ...
Bill Johnson's user avatar

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