**1**

vote

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**3**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**5**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**1**answer

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

**2**answers

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

**0**answers

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: ...