All Questions
9,780 questions
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147
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Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
- 121
9
votes
1
answer
611
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opposite Banach space
I heard this from Haskell Rosenthal many years ago.
If V is a complex vector space, say the opposite of V is the complex vector space with the same elements, the same operations except switch scalar ...
- 41.1k
3
votes
1
answer
332
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Continuity of a weight on its definition domain in a von Neumann algebra
Let $M$ be a von Neumann algebra and $\varphi$ be a normal weight on it,
and let $A$ be its definition subalgebra. We still denote $\varphi$
the extension to $A$ as a linear positive functional.
It ...
- 357
1
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0
answers
139
views
Can we define log-convex operators?
Let $I\subset\mathbb{R}$. A function $f:I\rightarrow\mathbb{R}$, is said to be log-convex if $\log f$ is convex or equivalently for all $x,y\in I$ and $\alpha\in [0,1]$
$$f(\alpha x+(1-\alpha)y)\leq [...
- 55
1
vote
1
answer
211
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Let $f \in M^{1,1} (\mathbb R)$ (Feichtinger's algebra /Modulation Space). Can we say $Fof\in M^{1,1}(\mathbb R)$; $F$ is an entire function?
The Modulation space ( Feichtinger's algebra),
$$S_{0} (\mathbb R) = M^{1, 1}(\mathbb R): = \{ f\in L^{2}(\mathbb R) : V_{g}(f) \in L^{1}(\mathbb R^{2}) \};$$
where $V_{g}f (x, w)$ is the short- ...
- 1,051
1
vote
2
answers
1k
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Convergence of eigenvectors
Let $T$ be a compact operator on $l^2$. Let $T_n$ be finite rank operators and $T_n \to T$ in the operator norm. Is it true that the eigenvalues and eigenvectors of $T_n$ converge to eigenvalues and ...
- 31
2
votes
0
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102
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Sobolev trace of $H^1(\mathcal{M} \times I)$ functions
Let $\mathcal{M}$ be a compact Riemannian manifold and let $I=(0,1)$. I seek a trace theorem saying that functions $u \in H^1(\mathcal{M} \times I)$ have a well-defined trace at $\mathcal{M} \times \{...
- 21
5
votes
2
answers
329
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Looking for substitutes for co-free modules in a topological setting
I should say that I'm not a category theorist or an abstract algebraist, so maybe this will be very pedestrian. I have the following, somewhat vague question:
I have categories C and D, a ...
- 18.7k
2
votes
0
answers
99
views
Equicontinuity of $\{f_{2n}\circ f_{2n-1}\}$
Let $(X,D)$ be a compact metric space and $\{f_n\}_{n\in\mathbb{N}}$ be a sequence of homeomorphisms of $(X,d)$. It is easy to see that if $\{f_n\}$ is uniformly convergent then $\{g_n\}$ defined by $...
4
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1
answer
2k
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Characterizations of a linear subspace associated with Fourier series
Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace
a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable ...
- 744
-1
votes
1
answer
259
views
Absolute continuity of probabilities on Polish spaces and open sets. [closed]
On a polish space $\mathcal{X}$ i consider two Borel probabilities $P$ and $Q$ such that for any open set $E$ of $\mathcal{X}$ we have : $P(E) =0$ implies $Q(E)=0$. Does this imply that $Q$ is ...
- 41
2
votes
2
answers
867
views
Decomposition of an abelian von Neumann algebra
Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance.
"Let $\pi$ be a non-degenerate ...
0
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0
answers
148
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existence of locally translation-invariant Borel measure on Frechet manifolds
It is well known that the only locally finite, translation-invariant Borel measure on an
infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
3
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0
answers
193
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Method to Generate Random Mutually Orthogonal Unitary Matrices
The title says it all, I'd like to know if such a method exists to generate random mutually orthogonal unitary matrices. If so, any supplementary references would be very much obliged. Thanks again!
- 133
0
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0
answers
184
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Can I define Fredholm Index using $\dim \ker ST - \dim \ker TS$?
$X$, $Y$ are Banach spaces.
Let $S \in L(X, Y)$, $T \in L(Y, X)$, where $L(X, Y)$ denotes the Banach algebra of bounded linear operators from $X$ to $Y$. If we have that $Id_Y - ST \in \mathbb{K}(Y)$ ...
- 365
3
votes
1
answer
133
views
A recurrent sequence related to the Brouwer fixed-point theorem
Let $K$ be a non-empty compact convex subset of a Banach space $E$, and let $f : K \longmapsto K$ be a continuous function. Fix $u_0 \in K$, and define by recurrence $u_{n+1} = \frac{1}{n+1} \sum_{j=0}...
- 7,249
1
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0
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295
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Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?
I first specify the setting and then formulate the question precisely. (A very long post follows.)
Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...
- 3,584
0
votes
1
answer
483
views
Absolute norms and 1-unconditional sums
Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, \|...
- 3
6
votes
0
answers
411
views
Birth-Death Process associated with Orthogonal Polynomials
I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...
- 22.8k
3
votes
0
answers
329
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Trace Norm Inequality for the Discrete Fourier Transform
I am having some trouble proving an inequality involving the trace norm or the operator $f{\cal{F}}_Ng$ where $f, g$ are diagonal matrices, f is positive semidefinite and $\cal{F}_N$ is the Discrete ...
- 141
9
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0
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885
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Continuous projections in $\ell_1$ with norm $>1$
I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case $...
- 1,697
0
votes
0
answers
145
views
Does there exist this special kind of homeomorphism?
Let $A,B\subset\mathbb{R}^n, n\geq 2,$ are two different shaped spindles. One is thick and one is thin. (Sorry for my unprofessional statements. Unsure about how to say it rigorously.) So there are ...
- 183
0
votes
0
answers
117
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The functional $L(\varphi)=\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$
Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$.
Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$.
Let $G\triangleq\{\varphi\in L^2[...
- 139
1
vote
2
answers
522
views
Variational problems whose lagrangian density depends on derivatives higher than 1.
The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
- 469
8
votes
1
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713
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Factoring operators $L_\infty \longrightarrow L_2$ as the composition of $n$ strictly singular operators, $n\in \mathbb{N}$
Motivation and background This question is motivated by the problem of classifying the (two-sided) closed ideals of the Banach algebra $\mathcal{B}(L_\infty)$ of all (bounded, linear) operators on $L_\...
- 2,378
2
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1
answer
431
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Sobolev imbedding
It is known that, for $n \ge 3, 2 < p< 2^*$, the imbedding $H^1(\mathbb{R}^n) \hookrightarrow L^p(\mathbb{R}^n)$ is not compact. Let $G=O(n_1) \times O(n_2)\times\cdots\times O(n_k)$, with
$n_1+...
- 397
2
votes
1
answer
139
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Reynolds operator from the potential theoretic point of view
In the book "Conditional Measures and Applications", it was pointed out that "Reynolds operators have not yet been studied from the potential theoretic point of view ." Have there been any research ...
- 21
2
votes
1
answer
901
views
Geometry of the Hilbert sphere
Let $X$ be the unit sphere in $\ell^2$, i.e. $X=\{x\in\ell^2: \|x\|=1\}$. Let the metric on $X$ be the geodesic metric, i.e. $d(x,y)=\cos^{-1}\langle x,y\rangle$. Call a set a ball-intersection if ...
- 744
2
votes
1
answer
179
views
Second quantization of partial isometry
If we have a unitary map from Hilbert space $H$ to $H$, we get a unitary map from $e^{H}$ to
$e^{H}$, where $e^{H}$ is the symmetric Fock space of $H$. But if we replace the unitary with partial ...
- 95
1
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2
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409
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Does these commutator estimates bound in $L^{2}$
According to the basic rules of symbolic caculus,$[a(x,D),x_{j}]=-ia^{j}[x,D]$.So we have $[(1-\triangle)^{\frac{1}{2}},x_i]=\partial_i(1-\triangle)^{-\frac{1}{2}}$ which is $L^2$ bounded.
It's also ...
- 1,644
0
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0
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107
views
Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?
Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$,
$$f(v) = A[v]_+ + B[-v]_+$$
surjective? Here $[.]_+$ is an elementwise ...
- 123
3
votes
2
answers
461
views
Complemented subspaces isomorphic to $c_0$ in $\mathcal{B}(E)$ [closed]
It is well known that neither
1) $c_0$ is isomorphic to a complemented subspace of $\mathcal{B}(H)$
nor
2) $c_0$ is a quotient of $\mathcal{B}(H)$
for a Hilbert space $H$. Can we replace $H$ above ...
0
votes
0
answers
137
views
SVD of Frechet derivative
This is mainly a reference request. Is there a particular characterization of operators A from a Hilbert space H to itself such that the Frechet derivative A'(u) exists for each $u \in H$ and for any ...
0
votes
1
answer
864
views
Sequence of smooth functions converging to sgn(x)
I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property:
\begin{equation}
f_i(x_1+x_2) = g_i(x_1, f_i(x_2))
\end{...
- 561
1
vote
1
answer
426
views
Continuous embedding of Hardy space in Lebesgue space
I would like to have a reference to the following statement which I think is true:
$$h^1 \hookrightarrow L^1.$$
The closest I came to this is in D. Goldberg's paper, "A local version of real Hardy ...
3
votes
1
answer
515
views
How to extend evaluation at a point from continuous maps to square-integrable ones?
Consider the Hilbert space $L^2[0, 1]$ of square integrable functions on $[0, 1]$.
Saying more carefully, there is one subtle detail: this space is defined as factor space by the functions which are ...
- 24.9k
5
votes
1
answer
331
views
Entire calculus and clmc algebras
If $\mathcal{A}$ is a complete locally convex (Hausdorff) associative unital algebra (over $\mathbb{C}$) one is interested in defining "transcendental" functions of a given algebra element $a \in \...
- 8,098
4
votes
1
answer
568
views
Crossed product of a non unital C*-algebra
Let $X$ be a locally compact space, and let $T:X\rightarrow X$ be a homeomorphism. Then \begin{align*}
&\alpha:C_0(X)\rightarrow C_0(X)\\\
&\alpha(f)=f\circ T
\end{align*}
is an automorphism. ...
- 43
1
vote
1
answer
283
views
$L^2$ boundeness of a sequence
Let $f_n \in C^2(\bar{\Omega})$ be a sequence satisfying
$\Delta f_n - f_n^3 \to 0 \ \ {\rm in} \ \ L^2(\Omega)$
where $\Omega \subset {\mathbb R}^2$ is bounded and open with a smooth boundary. Is ...
- 595
1
vote
1
answer
1k
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Laplace equation over concentric spheres
Is there a closed formula for the solution of Dirichlet problem ($\Delta u=0$) for annulus $r <|x| < R$, $x \in R^n$ (n>2), with two given boundary value functions, $f$ over $|x|=r$ and $g$ over ...
- 39
0
votes
1
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337
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Integral inequality
Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...
- 295
2
votes
1
answer
132
views
Form of finite dimensional contractive projection in $L_p$
Let $P$ be a finite dimensional contractive (norm 1) projection in $L_p$, $1 < p < \infty$. Then $P$ is of the following form:
$Pf = \sum_{k=1}^n g_k \int h_kf$
Where $\|g_k\|_p = \|h_k\|_q = \...
- 311
0
votes
1
answer
1k
views
Precompact set in L2 space?
Let A be a bounded interval in R. Suppose we have a collection of functions, such that
Each function is $\in$ $L^r(A)$, where r is any number $\in$ $[1, \infty]$,
The fractional derivative of order ...
- 11
3
votes
1
answer
375
views
Weak L_1-convergence of squares
Let $f_k$ be a sequence of non-negative functions from $L_2(\Omega)$, where $\Omega$ is a bounded open set. Assume that $f_k\to f$ weakly in $L_2$ and strongly in $L_p$, $\forall p<2$. Assume also ...
- 43
0
votes
1
answer
220
views
Spectral decomposition function [closed]
Once I met a notation of "spectral decomposition function" (for a self-adjoint operator). No definition was given.
Could someone give me a clue what can that be, cause I can't find this exact phrase ...
- 1
1
vote
0
answers
136
views
A linear operator equation (PDE) with non-monotone term
I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...
- 11
2
votes
0
answers
98
views
What does integrability of a strictly monotonic function imply about the tails of that function?
In particular, if $f:\mathbb{R}_{+}\rightarrow[0,1]$ is a strictly monotonic decreasing function and $f$ is integrable then does it necessarily hold that $f^{-1}(1/t)=o(t)$?
1
vote
0
answers
170
views
A question about Smulian lemma
Smulian lemma says Let $(X, ||.||)$ be a Banach space and$(X^*, ||.||^*)$ and let $x\in S_X=\{x\in X:||x||=1\}$ then
(i) $||.||$ is Frechet diffrentiable at $x$ iff $\lim\limits_{n\to\infty}||f_n-...
- 823
0
votes
0
answers
206
views
About approximate eigenvalue
I am in trouble when read the book "D.Henry, Geometric Theory of Semiliner Parabolic Equations". The question is relate to Page 104,proof Lemma 5.1.4.
Suppose $X$ is a real Banach Space, $M$ is a ...
- 103
6
votes
2
answers
293
views
quasinilpotence and finite spectrum
Let A be a quasinilpotent operator on a Hilbert space and let $A^{*}A$ have finite spectrum.
Does then follow, that A is nilpotent ?
- 2,753