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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Semigroups of matrices closed under conjugate transposition

An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
E W H Lee's user avatar
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Monotonicity of log determinant of Gaussian kernel matrix

Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation} be a squared-exponential (Gaussian) kernel, with $\sigma,\vartheta>0$. Consider, for a set of $N$ ...
Heinrich A's user avatar
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251 views

Struggling with a proof in the preprint 'Hermitian geometry on resolvent set'

I have been struggling for awhile with a particular argument in the paper below. I posted the question first on MathSE, but I got no answers. I understand however that MO might be an overreach for ...
WeakMath's user avatar
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258 views

Shift on trivalent directed tree, operator and von Neumann algebra

Let $\mathcal{T}$ be the trivalent directed tree, with two parents and one child for each vertex (see below). Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$ and $H$ be the Hilbert space $\...
Sebastien Palcoux's user avatar
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280 views

Lax pairs in an abstract formalism

I am reading Integrals of Nonlinear Equations of Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide ...
Delio Mugnolo's user avatar
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298 views

Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? : $V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). $...
Keith McClary's user avatar
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About generator and isomorphism problems for free groups operator algebras

Let $H$ be an infinite dimensional separable Hilbert space. The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$. Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
Sebastien Palcoux's user avatar
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2 answers
627 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
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7 votes
3 answers
832 views

Criterion for compactness

Let $T:H\to H$ be a continuous operator on a Hilbert space. Assume there exists an orthonormal base $(e_j)_{j\in\mathbb N}$, such that the sequence $Te_j$ tends to zero. Must $T$ be compact?
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Is this a characterization of commutative $C^{*}$ algebras?

Assume that $A$ is a $C^{*}$ algebra with self adjoint elements $A_{sa}$. Assume that for all $a,b\in A$ we have $$ab\in A_{sa} \iff ba \in A_{sa}$$ Is $A$ necessarily a commutative ...
Ali Taghavi's user avatar
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Ramanujan's Master Formula: A proof and relation to umbral calculus

The Ramanujan's master theorem states that: $$ \int_0^{\infty}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^n}{n!}a_nx^ndx=\Gamma(s)a_{-s} $$ I found a really strange proof recently on a personal blog: Define $...
FFjet's user avatar
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3 answers
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Commutant of the conjugations by unitary matrices

Let $\mathcal{L}(\mathbb{C}^{n \times n})$ denote the algebra of all linear mappings from $\mathbb{C}^{n \times n}$ to $\mathbb{C}^{n \times n}$ and let $\mathcal{C} \subseteq \mathcal{L}(\mathbb{C}^{...
Jochen Glueck's user avatar
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2 answers
538 views

Existence of spectral gap

I would like to start by saying that any comment or idea is highly appreciated. Let us observe that for Hilbert-Schmidt operators $H_1,H_2$ on an infinite-dimensional separable complex Hilbert space $...
Dixmier's user avatar
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3 answers
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Simultaneous diagonalization of self-adjoint operators on Hilbert space

Suppose I'm given a finite set of possibly unbounded commuting self-adjoint operators $T_i : \mathfrak H \supset \mathscr D(T_i)\to \mathfrak H, i = 1 , \dots , N$ on a Hilbert space (in the sense of ...
jacques's user avatar
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4 answers
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On operator ranges in Hilbert & Banach spaces

Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent: (1) ran($A$) $\subset$ ...
Tom LaGatta's user avatar
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Spectrum of "classical" operators

Lately, I've been reading a couple of papers from different one-dimensional PDE contexts on which operators like $\mathcal{L}:=-\partial_x^2+c_*+\Phi$ repeatedly appear. Usually, on these contexts $\...
Sharik's user avatar
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2 answers
428 views

Non strictly-singular operators and complemented subspaces

If $T$ is a bounded operator which is not strictly singular, acting on a separable Banach space $X$, can one always find an infinite dimensional, closed and complemented, subspace $Y$ such that $T$ ...
Markus's user avatar
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3 answers
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Essential spectrum of multiplication operator

Let $a\in \mathcal{L}(L^2([0, 1], \mathbb{R}))$ be a multiplication operator. I wonder whether there is any work on calculating its essential spectrum. Is there any way to explicitly compute its ...
potionowner's user avatar
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1 answer
276 views

Does separability of the strong operator topology imply separability of the underlying space?

Let $X$ be a Banach space and $B(X)$ be the space of bounded operators on $X$. Suppose that the strong operator topology on $B(X)$ is separable and that the cardinal number of $B(X)$ is continuum. ...
ABB's user avatar
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1 answer
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What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an $\mathbb{R}$-...
alexlo's user avatar
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1 answer
240 views

Is there a nice "minimum" of two symmetric operators?

Let $A$ and $B$ be two bounded symmetric positive operators in Hilbert space, such that $A-B$ is trace class. If needed, $A$ and $B$ may be assumed reasonably "small", let's say, Hilbert-Schmidt. ...
Alexander Shamov's user avatar
7 votes
1 answer
183 views

numerical radius of hyponormal operator

Let $F$ be a complex Hilbert space. We recall that an operator $S\in\mathcal{B}(F)$ is said to be hyponormal if $S^*S\geq SS^*$ (i.e. $\langle (S^*S-SS^*)z,z \rangle\geq 0$ for all $z\in F$). ...
Schüler's user avatar
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1 answer
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Convergence of $\exp(tQ)$ in operator norm as $t\rightarrow\infty$

This is not a homework problem, so I am not sure whether this has a "good" answer or not. I came up with this question when I am now learning functional analysis and wonder whether my "...
MikeG's user avatar
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1 answer
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Is every sequentially $\sigma(E',E)$-continuous linear functional on a dual Banach space $E'$ necessarily a point evaluation?

$\newcommand{\bf}[1]{\mathbb #1}\newcommand{\sc}[1]{\mathscr #1}$ A duality between two vector spaces $E$ and $F$ over $\bf K$ ($= {\bf R}$ of ${\bf C}$) is, by definition, a bilinear form $$ \...
Ruy's user avatar
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1 answer
531 views

Fractional powers of an operator

What is the large class of operators for which one can define fractional powers? For example, we can consider an operator $A: D(A) \subset X \rightarrow X$, generator of an analytic semigroup on a ...
Migalobe's user avatar
  • 395
7 votes
1 answer
427 views

Extending compact operators

Let $X$ be a separable, infinite-dimensional complex Banach space and $Y\subseteq X$ an infinite-dimensional closed subspace. Suppose $K:Y\to X$ is an arbitrary compact operator. I would like to ...
Ben W's user avatar
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1 answer
220 views

A characterization of Hilbert spaces by norm one projections

Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert ...
Markus's user avatar
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2 answers
461 views

Extension of weakly compact operators from $\ell_1$ into $c_0$

Is every weakly compact operator from $\ell_1$ into $c_0$ extendible to any larger space? Equivalently, is every weakly compact operator from $\ell_1$ into $c_0$ extendible to $\ell_\infty$?
Joaquin M. Gutierrez's user avatar
7 votes
1 answer
227 views

Is a sign-preserving operator on $L^2$ a multiplication?

Let $T:L^2(\mu)\to L^2(\mu)$ be a linear and continuous operator, where $L^2(\mu)$ is the (real) $L^2$-space to some $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$. $T$ is assumed to be sign-...
daw's user avatar
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1 answer
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Property between trace class and Hilbert-Schmidt

Consider the following condition on a bounded operator $T$ on a Hilbert space: $\ \ \ \ \ $(A) there exists an orthonormal basis $(e_j)$ with $\sum_j\parallel Te_j\parallel<\infty$. We have the ...
user avatar
7 votes
1 answer
324 views

Is every $C_0$ semigroup on a Hilbert space automatically a $C_0$ group on a larger space?

Let $\{T(t),t\ge 0\}$ be a $C_0$ semigroup on a Hilbert space $X$, does that exist a larger Hilbert space $Y$ such that $X\subset Y$, and $T(t)$ extend to a $C_0$ group $T'(t)$ (so $t<0$ make ...
Tomas's user avatar
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7 votes
1 answer
423 views

Gradient flows on Hilbert manifolds

I would like to know if gradient flows of Morse-Bott functions on a Riemannian manifold always converge towards a unique critical point, provided that the flow line is bounded. To be more precise, a ...
Schlubbidubbi's user avatar
7 votes
1 answer
322 views

Why is the definition of von Neumann trace independent of the choice of the Hilbert space?

A Hilbert module defined in "L^2-invariants: theory and applications to geometry and K-theory", Springer-Verlag, 2002, by W. Lück: A Hilbert $\mathcal N(G)$-module $V$ is a Hilbert space $V$ ...
Meisam Soleimani Malekan's user avatar
7 votes
1 answer
741 views

An equivalent condition for separability of $X^*$

Let $X$ be a Banach space. By the weak operator topology on $B(X)$, we mean the locally convex topology implemented by the following semi-norms: $$B(X)\to[0,\infty) : T\to|\langle Tx,x^*\rangle|$$ ...
ABB's user avatar
  • 3,898
7 votes
1 answer
353 views

Criteria for operators to have infinitely many eigenvalues

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem. For non-normal operators this no longer has to be true. There ...
Sascha's user avatar
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7 votes
1 answer
261 views

Simple $C^*$ algebras with invariant subspace property

Edit: According to the valuable comment of Yemon Choi I revise the question by replacing "faithful" with "irreducible". We say that a $C^*$ algebra $A$ satisfies the invariant subspace ...
Ali Taghavi's user avatar
7 votes
1 answer
306 views

An indicator of a planar subset as an element of a tensor product

Denote $I=(0, 1)$, and let $\mu$ be the Lebesgue measure on $I$. Does there exist a function $f$ on $I\times I$ viewed as an element of the space $L^\infty(\mu\times\mu)$ such that $$ f^2=f $$ (that ...
limanac's user avatar
  • 452
7 votes
1 answer
379 views

Why are we interested in operators that share a basis of eigenfunctions?

I hope this is an appropriate question for this forum. If not, I apologize. Before stating my question (which may be found at the end of this post), I will attempt to provide sufficient context. I ...
Quoka's user avatar
  • 133
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0 answers
149 views

What is the current status of research on the von Neumann's inequality for $n \ge 3$?

Problem Let $(T_1, \ldots, T_n)$ be a tuple of commuting contractions in Hilbert space $H$. Does a constant $C_n \ge 1$ exist, for which it would be true, that: $$\forall_{p \in \mathbb{C}[x_1, \ldots,...
S-F's user avatar
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0 answers
167 views

Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements

Consider the free product of $\mathbb{Z}/2$ with itself with generators $$ \mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle $$ and regard its group $C^*$-algebra $$ C^*(\mathbb{Z}/2*\mathbb{...
C-star-W-star's user avatar
7 votes
0 answers
196 views

Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
geometricK's user avatar
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7 votes
0 answers
83 views

Analogue of Friedrichs extension for Hilbert $C^*$-modules

Suppose one has a densely defined symmetric operator $T:\mathcal{M}\rightarrow\mathcal{M}$, where $\mathcal{M}$ is a Hilbert $A$-module for a $C^*$-algebra $A$. Suppose that $T$ is non-negative, so ...
geometricK's user avatar
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7 votes
0 answers
180 views

A single vertex operator - bare bones explanation?

There is an ocean of literature (and a sea of popular texts inside) on vertex algebras, including quite a lot of Q & A here on MO, and I am trying to read some random selections from time to time. ...
მამუკა ჯიბლაძე's user avatar
7 votes
0 answers
364 views

Spectra of Dirac operators

1. Suppose that $M$ is a spin manifold. The spin structure of $M$ is not uniquely defined: in other words, $M$ may have many nonequivalent spin structures. For each choice of spin structure there is ...
seksi_tosia_69's user avatar
7 votes
0 answers
178 views

Is there a quotient of $c_0$ without the approximation property?

The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property? This would follow if ...
user136256's user avatar
6 votes
3 answers
439 views

Norm estimation of identity plus two non-commuting self-adjoint operators

This is a problem that I have been stuck for a few months. Let $X$ be a Hilbert space and $A:B:X\to X$ be two non-commuting semi-positive self-adjoint bounded linear operators. Is it true that $$\|(I+...
John's user avatar
  • 483
6 votes
3 answers
845 views

Are nuclear operators closed under extensions?

Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram $\require{AMScd}$ \begin{CD} 0 @>>> X_1 @>f_1>> X_2 ...
santker heboln's user avatar
6 votes
3 answers
956 views

Invertible operator with countable spectrum

Let $H$ be a separable Hilbert space and $A$ is an invertible bounded operator on $H$. Can we approximate $A$ with an invertible operator $B$ such that $sp(B)$ is a countable set? Motivation: ...
Ali Taghavi's user avatar
6 votes
2 answers
596 views

Is the injective envelope functorial?

Let $A$ and $B$ be unital $C^*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^*$-...
Andromeda's user avatar
  • 189
6 votes
3 answers
2k views

Non-empty resolvent set, then operator closed?

On Hilbert spaces, the following is true: Let $T$ be a densely-defined linear operator with non-empty resolvent set, then $T$ is closed. The obvious proof I see to show this uses explicitly the ...
gipom's user avatar
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