Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

learn more… | top users | synonyms (1)

1
vote
0answers
74 views

Convolution Integral involving an unknown function

I've got the following problem I'm working on which is related to some of my research. I am trying to solve the following equation for the function $f$. $$t^{-\alpha} \exp{ \left(- \beta x^2 t^{-2 ...
0
votes
0answers
55 views

What equals $\ker[(A-\lambda I)^+]$ for a negative unbounded operator $A$?

We have the following result: $\{ E_{\lambda}; \, -\infty <\lambda < + \infty\}$ is a spectral family, where $E_{\lambda}$ is the projection of $H$ onto the null space $\mathscr N ...
1
vote
0answers
42 views

Multilinear Interpolation

Suppose I have a multilinear map $\Gamma(u,v)$ satisfying \begin{align} \big\| \Gamma(u,v)\big\|_{L^2} &\leq \big\| u\big\|_{L^2} \big\| v\big\|_{L^2} \\ \big\| \Gamma(u,v)\big\|_{L^\infty} ...
2
votes
1answer
101 views

Uniqueness from orthogonality relation?

This question was posted yesterday on MathOverflow by Michael Smith and received a number of upvotes. I too think the question was interesting. However, for some unknown to me reasons, it has been ...
2
votes
0answers
63 views

Getting an a priori energy estimate from PDE weak formulation

On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying $$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int ...
1
vote
1answer
204 views

On the second dual of $C[0,1]$

I have two questions on the second dual of $C[0,1]$: R. D. Mauldin ([1]) proved that: For a given bounded linear functional $T: C[0,1]^*\to \mathbb{C}$ there is a bounded function $\psi$ defined on ...
1
vote
3answers
112 views

Norm of an operator formed using a unitary operator

Suppose, $ A $ is a unitary matrix in $ M_n(\mathbb{C}) $ given by $ (a_{i,j})_{1\le i,j\le n} $ which has the property that, for all the basis elements $ e_i $, $ Ae_i\ne |\lambda| e_j $ for all $i,j ...
2
votes
0answers
83 views

Are Ritt operators mean ergodic?

In the following, $T$ is a bounded operator on a Banach space $X$. $T$ is called "power bounded" if $\sup_{n\in \mathbb N}\|T^n\|<\infty$; $T$ is called "mean ergodic" if the Cesàro sums ...
2
votes
0answers
108 views

The boundedness of an entire function along the imaginary axis

I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...
1
vote
0answers
30 views

Boundary regularity of higher order PDE

consider the subsequent pde (weak formulation): $\int_\Omega D^m\phi:D^m\psi+ Df(D\phi):D\psi+(g h\circ\phi)\cdot\psi dx=0$. In this case, $n\geq 2$, $\Omega=[0,1]^n$, $m>2+\frac{n}{2}$, ...
8
votes
5answers
420 views

Applications of functional analysis beyond analysis(towards algebra, geometry, number theory…) [closed]

So far, We have seen the applications of functional analysis in PDE, probability and many areas in applied mathematics. On the other hand, methods of algebraic topology are introduced to functional ...
0
votes
0answers
57 views

Topologies with the same convex closed sets

Let $\tau_1$ and $\tau_2$ be locally convex Hausdorff topologies on vector space $X$ such that $(X,\tau_1)^\ast = (X,\tau_2)^\ast$. It is well known that $(X,\tau_1)$ and $(X,\tau_2)$ have the same ...
2
votes
0answers
46 views

When does the ground state energy continuously depend on a parameter?

Given a family of Schrödinger operators $H_\gamma=-\Delta+V_\gamma$, under which condition is the map $\gamma\mapsto\inf\sigma(H_\gamma)$ continuous? This is surely the case for many textbook ...
2
votes
0answers
146 views

Is this continuous linear map weakly compact?

Let $E$ and $F$ be Fréchet spaces, let $U$ be an open subset of $E$, and let ${\mathcal{H}}(U;F)$ be the spaces of holomorphic mappings from $U$ into $F$. Let $\tau_c$ denote the compact-open ...
1
vote
0answers
53 views

Properties of convergence at points of continuity

Let $J$ denote the set of functions $f : [0, \infty) \to \mathbb{R}$ that are right-continuous and have left-hand limits (r.c.l.l.) and such that their points of discontinuity are jumps. Then $J$ is a ...
8
votes
3answers
472 views

is every element in a C* algebra a product of normal elements?

I have the following question and since I am not an expert on C*-algebras, I thought I ask it here: I know that in general the sum and product of normal elements need not be normal. It is even true ...
1
vote
1answer
53 views

Extremal of an L^1 continuous functional on a compact bounded set

Please, I need a small help with a reference. Lets say we do have a continuous functional $f$ on $L^1$ space and we want to prove the existence of extremals $f(\Omega)$, where $\Omega$ is compact and ...
3
votes
0answers
138 views

A strongly open set which is not measurable in the weak operator topology

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$. Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: ...
1
vote
1answer
69 views

Definite intergal with two K-Bessel functions and x

I would like to calculate the definite integral with K-Bessel funcitons and a and b complex (n and k integers): $$\int_{0}^{\infty} x \;K_{a}(nx) \; K_{b}(kx) \; dx$$ I could not find it in ...
0
votes
1answer
118 views

A point-wise separation Hahn-Banach theorem in C*-algebras

Let $H$ be a Hilbert space. We denote $K(H)$ by the space of compact operators on $H$ which is a two sided ideal in $B(H)$. Let $E$ be a norm closed convex subset of positive operators in $K(H)$ ...
4
votes
0answers
54 views

L^1 maximal inequalities for the Ornstein-Uhlenbeck semigroup in infinite dimension

For an infinite-dimensional Gaussian random vector $X$ consider the Ornstein-Uhlenbeck maximal operator: $M f(X) := \sup_{\rho \in [0,1]} \mathsf{E} [f(\rho X + (1-\rho^2)^{1/2} X^\prime) \mid X]$ ...
5
votes
2answers
150 views

$L^{\infty}$ polynomial approximation

In short: For a given smooth or continuous function, how can we obtain the best $L^{\infty }$ approximating polynomial? Jackson (1911) proved that there is a best approximating polynomial in the ...
1
vote
0answers
72 views

Constructing special holomorphic functions

I have thought about this for a while and I can not get a suitable way of proceeding, so any help is appreciated. Suppose $D$ is the unit disk. And $f$ is a continuously differentiable strictly ...
2
votes
0answers
41 views

Functional equations about Conway's box function

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). The ...
2
votes
0answers
37 views

Restricted weak type bound at the endpoint

We know that if we have an operator that is (restricted) weak type $(p,p)$ and (strong) type $(\infty,\infty)$ with norm 1, then it's also of strong type $(q,q)$ for all $p<q<\infty$ by the real ...
0
votes
0answers
81 views

The norm of the operator in the Calderon-Marcinkiewicz interpolation theorem

I read a general Marcinkiewicz interpolation theorem (the Calderon-Marcinkiewicz theorem) in J.Bergh's book "Interpolation Spaces - An Introduction".(Page 113-114, Theorem 5.3.2). If $T:L_{p_ir_i}\to ...
2
votes
0answers
184 views

Quantum Mechanics derivation of Wallis' Formula?

Recently there was a proof of the Wallis Product using quantum mechanics on the arXiv. However, there are many proofs of the result, Wikipedia has 4. Fine Print the first proof has on Wikipedia, the ...
12
votes
2answers
861 views

Do non-stable Banach spaces exist?

Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties: Is every infinite ...
2
votes
0answers
30 views

Specific type operators and basic sequences

Let $s$ be the space of rapidly decreasing sequences, i.e. $s=\{\xi=(\xi_j)_j\colon\,\,\sup_j|\xi_j|j^n<\infty\,\,\text{for all}\,\,n\in\mathbb{N}\}$ and $s'$ its topological dual, i.e. ...
2
votes
1answer
74 views

Strongly continuous semigroups and symbols of pseudo differential operators

I am considering the Cauchy IVP for the evolution equation $$u_t + \Psi u =0$$ where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$. The ...
6
votes
1answer
110 views

Convex Hull of univalent functions and Bieberbach Conjecture

Consider the class $S$ of univalent functions on the unit disk $D$ normalized so that $f(0)=0$ and $f'(0)=1$. Each function in $S$ satisfy the Bieberbach conjecture, that is the $n$-th coefficient in ...
1
vote
1answer
99 views

Nonlinear elliptic problem involving the p-laplacian, Hölder inequality

I am studying the paper On some nonlinear elliptic problems for $p$-Laplacian in $\mathbb{R}^n$ by Abdelouahed El Khalil and Said El Manouni Mohammed Ouanan. I have a problem understanding one step ...
9
votes
2answers
257 views

Nuclear operators/spaces and transfer operators

While studying for my thesis (in dynamical systems) I've encountered multiple times with the concept of nuclear operators and nuclear spaces, often linked with the works of Grothendieck. For example, ...
7
votes
1answer
196 views

Tightness and Functional Analysis

Let $(\Omega , \mathbb{P})$ be a probability space and $X$ be a real-valued random variable. Then we immediately have the push-forward measure $\mu$ on $\mathbb{R}$ and one can think of $\mu$ as an ...
4
votes
0answers
38 views

Chord-arc property of n-tuples of commuting operators

Assume that we have an $n$-tuple $S^0=(S^0_1,\dots,S^0_n)$ of commuting operators in a Hilbert space $H$ and another such an $n$-tuple $S^1$. Is it possible to connect these two $n$-tuples by a ...
3
votes
1answer
124 views

A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$

I asked this at math.stackexchange, but nobody answered. Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ...
4
votes
1answer
171 views

Application of Factorization Theory to Oscillatory Integral Estimates

In the article "Some New Estimates on Oscillatory Integrals" by Bourgain in the book Essays in Honor of Elias M. Stein, Bourgain considers operators of the form ...
1
vote
0answers
52 views

Uniform limit of m-homogeneous polynomials over compact subsets of a Banach Space

I am trying to solve problem 1.2.A from Mujica's book "Complex Analysis in Banach Spaces". We denote by $\mathcal{P}_a(^mE;F)$ the space of all $m-$homogeneous polynomials from $E$ into $F$, i.e, the ...
3
votes
1answer
135 views

Hardy-Littlewood-Sobolev inequality in Lorentz spaces

Hardy-Littlewood-Sobolev inequality states that if $1<p<q<\infty$, $1/r=1-1/p+1/q$, then we have $$\left\|\frac{1}{|x|^{n/r}}\ast f\right\|_{L^q(\mathbb R^n)}\le\|f\|_{L^p(\mathbb R^n).}$$ ...
4
votes
2answers
182 views

On the possibility of extending the Sz.-Nagy dilation theorem for multiple contraction operators on Hilbert spaces

I am presently doing research concerned with operator algebras and operator theory and I thought to write here in the hopes of seeking expert advice on an idea I had here. The classic Sz.-Nagy ...
2
votes
0answers
151 views

Every convex sequentially closed set is closed

Let $X$ be a vector space. A vector (not necessarily Hausdorff) topology on $X$ will be called convex sequential if every convex sequentially closed subset of $X$ is closed. Is there some description ...
2
votes
2answers
134 views

Finding a specific Global Smooth Function

Any help with this problem would be appreciated. Thanks Suppose $(M^3,g)$ is a smooth compact Riemannian manifold with smooth boundary and $\gamma$ is a simple smooth orientable curve in $M$. Does ...
6
votes
1answer
158 views

von Neumann algebras as C*-algebras with multiplicative conditional expectation $A^{**}\to A$

Let $A$ be a C*-algebra. We identify $A$ with its canonical image in the bidual $A^{**}$. Consider the following conditions: (1) $A$ is a von Neumann algebra. (2) There is a multiplicative ...
3
votes
0answers
74 views

Automorphism group of Lie algebra of bounded operators

What is the automorphism group of the complex Lie algebra of bounded operators on a complex Hilbert space, with the commutator as Lie bracket? What for the real Lie algebra of bounded antihermitian ...
0
votes
0answers
36 views

regularity of the conditional expectation: from Markov to Non-Markov

Let $B=(B_t)_{0\le t\le T}$ be a standard Brownian motion and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Let $\xi=\xi(B)$ be a bounded measurable functional. Now let's ...
1
vote
2answers
214 views

Density of sets whose image is dense

This is probably easy, but I can't think of an answer. Assume $X$ is a Banach space and $A$ is a (not assumed closed) subspace of $X$. Let $T:X \to X$ be a bounded linear operator, which is also ...
5
votes
2answers
488 views

Holomorphy of a function with values in a Hilbert space

I asked the same question on MathStackExchange. EDIT: This question has now a open bounty worth +50 reputation on MSE. Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq ...
3
votes
1answer
105 views

Two minimization problems using singular value decomposition

Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition Let $q_0, q_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are ...
5
votes
2answers
172 views

Help in understanding result from publication on operator theory

in my research on dilations of contractions on Hilbert spaces and manifolds I have come across this nice publication concerning the classic Sz-Nagy theorem on the Arxiv by Levy and Shalit which states ...
1
vote
0answers
54 views

Finding the infimum of the range of a certain non-negative function associated to a $ C^{*} $-algebra

Let $ A $ be a non-trivial $ C^{*} $-algebra and $ n \in \mathbb{N} $. Setting $ \mathcal{D} \stackrel{\text{df}}{=} A^{n} \setminus \{ (0_{A},\ldots,0_{A}) \} $, we can define a function $ f: ...