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

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9
votes
3answers
184 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 ...
0
votes
2answers
101 views

Semi-reflexive dual

I am looking for an example of a semi-reflexive locally convex topological vector space, whose strong dual is not semi-reflexive. Is there some well-known example ?
0
votes
2answers
174 views

Isomorphism theorem for subfactors?

It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors : Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
16
votes
1answer
464 views

Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related?

It is known that there are only 14 reasonable tensor norms in $Ban$. On the other hand it is well known fact for topologists that one can obtain only 14 different sets from a given set applying ...
11
votes
2answers
357 views

Can non-central projections still commute with all other projections?

Let $A$ be a C*-algebra and let $\mathcal{P}(A)$ denote the set of projections in $A$. If $p\in\mathcal{P}(A)$ commutes with everything in $\mathcal{P}(A)$ does it necessarily commute with everything ...
0
votes
0answers
60 views

Open ideally convex sets

Background Recall that a subset $A \subseteq X$ of a Banach$^{1}$ space $X$ is said to be ideally convex if, for every bounded sequence $(x_n)_{n \in {\mathbb N}}$ in $A$ and every sequence ...
17
votes
2answers
698 views

Example of a space for which $V \cong Hom(V,V)$

Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology. Is there a non-trivial ...
7
votes
3answers
271 views

Rosenthal like inequality for weak $\mathbb L^p$-norms

Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if ...
4
votes
2answers
108 views

Minimax theorem on a non convex domain

A minimax theorem is a theorem which states that under certain conditions on $\mathcal{X}$, $\mathcal{Y}$ and $f$: $$ \inf_{x \in \mathcal{X}}{\sup_{y \in \mathcal{Y}}{f(x,y)}} = \sup_{y \in ...
4
votes
1answer
136 views

Examples of special isometries

Are there examples of (distinct) Hilbert spaces $H_1$=$(H,\langle\cdot,\cdot\rangle_1)$, $H_2 $=$(H,\langle\cdot,\cdot\rangle_2)$ and a linear operator $V: H_1\to H_2$ such that $V^n: H_1\to H_2$ is ...
12
votes
1answer
148 views

Can a non-commutative C*-algebra be a minimal operator space?

By an operator space structure on a Banach space $X$ I mean a sequence of norms on spaces $M_n \otimes X$ that satisfies Ruan's axioms. Among such admissible sequences there is always the smallest ...
-1
votes
1answer
110 views

Is the countably infinite product of locally convex topological vector spaces locally convex?

Let $(X,\tau)$ be a locally convex topological vector space and denote the product space $$X^{\infty}=X\times X\times X\cdots:=\big\{x=(x_i)_{i\geq 1}:~ x_i\in X\big\}$$ If we endow $X^{\infty}$ ...
4
votes
1answer
89 views

Does the group of compact perturbations of the identity act transitively on the compact operators?

Let $H$ be an infinite dimensional (separable if necessary) complex Hilbert space, and denote by $K(H)$ the ideal (in $B(H)$) of compact operators on $H$. Let $G_c=\{I+K\in B(H): I+K \text{ is ...
1
vote
1answer
162 views

Inequality in the Sobolev space $H^1$

I've found the following inequality $$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ...
4
votes
1answer
392 views

A generalization of a theorem of Grothendieck

In this question the norm of $L^{P}[0,1]$ is denoted by $\parallel . \parallel _{p}$. Let $p$ and $q$ be two arbitrary real numbers with $2<p<q$. Assume that $S$ is a subvector space ...
0
votes
1answer
115 views

Some convergence similar to weak-$\ast$ convergence on the space of finite measures

I have a question: Let $D$ be the space of cadlag functions defined on $[0,1]$ and $V$ be its subspace consisting of $x$ with finite variation and $x(0)=0$. Define $TV(x)$ as the total variation ...
0
votes
1answer
129 views

The dual space of the Dirac measures on an Abelian group

Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group. Question. What would be natural vector space $\mathcal{R}$ ...
8
votes
1answer
221 views

Grothendieck spaces and total subspaces of the dual

There is probably an embarrassingly simple counter-example to my question but I couldn't figure it out myself. Let me give it a try here. A Banach space $X$ is Grothendieck if weak*-convergent ...
1
vote
1answer
122 views

a question about the proof of identification of dual space

I have a question about the proof below: Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded ...
0
votes
3answers
168 views

The importance of basis constant in Banach spaces

Let $X$ be a Banach space an let $(e_n)_{n=1}^{\infty}$ be a Schauder basis of $X$. If we denote the sequence of the natural projections associated with $(e_n)_{n=1}^{\infty}$ by ...
2
votes
2answers
144 views

Perturbations of positive-definite self-adjoint operators

I was reading Kato's book on Perturbations of Linear Operators and have the following questions: If we have a self-adjoint operator, what kinds of perturbations (other than relatively bounded ones) ...
3
votes
0answers
104 views

Fourier transform and support of a distribution

Let $T \in \mathscr D'(\mathbb R^n)$ be a distribution, such that its Fourier transform $\widehat T$ is a real analytic function on $\mathbb R^n$ but it can't be continued to an entire function on ...
1
vote
1answer
129 views

Is the speed of a curve in $ \ell^\infty $ zero a.e. if the derivative of each component is zero a.e.?

Let $ A $ be an $ \mathcal{H}^1$-measurable subset of $ \mathbb{R} $ and $ \gamma \colon A \subseteq \mathbb{R} \to \ell^\infty $ be a Lipschitz mapping with the Lipschitz constant $ L $. Also, assume ...
4
votes
0answers
46 views

Strictly convex renormings making power bounded operators into contractions

Let $X$ be a Banach space and let $T$ be a power bounded linear operator on $X$ (i.e. $\sup_{n\ge0}\|T^n\|_{op}<\infty$). We can of course define an equivalent norm $\|\cdot\|'$ on $X$ so that ...
3
votes
1answer
143 views

(Ref req) Schrödinger heat kernel is a weak solution of parabolic Schrödinger equation

If we have nonnegative $V \in L^1_{\textrm{loc}}(\mathbb{R}^{n})$, then the operator $H = -\Delta + V$ can be defined on $L^{2}(\mathbb{R}^{n})$ via quadratic form methods. This is done by, for ...
0
votes
0answers
87 views

Convergence of functions of bounded variation

Let $u_n$ be a sequence of smooth functions with $||u_n||_{W^{1,1}}<C<\infty$. By $BV$ compactness, a subsequence of $u_n$ converges to some $u$ in $L^{n/(n-1)}$. Assume $\int_{\Omega} (|\nabla ...
1
vote
0answers
66 views

equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)

Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$, $$ ...
0
votes
0answers
64 views

Harmonic/functional analysis question: Uniform bounds for $(1 - \varepsilon\Delta)^{-1}$ as $\varepsilon\to 0$?

Is there a space $X$, compactly embedded in $H^{-1}(R^2)$, such that the operators $(1 - \varepsilon \Delta)^{-1}$ are bounded from $L^1(R^2)$ into $X$, with operator norms that are in turn bounded by ...
2
votes
1answer
294 views

Is every distribution a linear combination of Dirac deltas?

My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space $\mathcal{S}(G)^\times$ of tempered distributions on $G$, so that any distribution ...
4
votes
2answers
228 views

Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces

This question was posed on MathStackExchange but did not get an answer (even with a bounty). In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert ...
3
votes
2answers
107 views

series representation of bivariate functions

Given a bivariate function $f(x, y)$ with $x \in [-a,a]$ and $y \in [-b, b]$, what is the necessary and sufficient condition under which we can write $f(x, y) = \sum g_k(x)h_k(y)$ for all $(x,y)$ in ...
5
votes
1answer
343 views

Do exist infinitely differentiable, compactly supported non zero solutions of the free Schrodinger equation?

I would like to get an answer for the following problem (and possibly be pointed to the relevant literature): given the one dimensional free Schrodinger equation $ i \, f_t + f_{xx}/2 = 0$ for the ...
2
votes
1answer
71 views

Support-preserving pseudodifferential operators

Let $A = F^{-1}\sigma F$ be a pseudodifferential operator acting on functions on $\mathbb R^n$, where $F$, $F^{-1}$ are the direct and inverse Fourier transforms respectively and $\sigma$ is the ...
1
vote
0answers
88 views

Domain of the adjoint of the Laplacian

Given a domain $\Omega \subset R^n$, consider the set $D := \{u \in L^2(\Omega)| \Delta u \in L^2(\Omega)\}$, where $-\Delta$ is the Laplacian. I think this is the domain of the adjoint of $-\Delta$. ...
0
votes
1answer
128 views

A Poincaré-type inequality with logarithmic function

For any function $f(x)$ we denote $\bar{f}:=\frac{1}{\Omega}\int_\Omega f(x)\,dx$. Let $\Omega\subset \mathbb{R}^n$ be a bounded smooth domain and $u(x)> 0$ be a smooth function defined on ...
8
votes
0answers
228 views

Carleson's Theorem on Manifolds

Let $M$ be an oriented, compact, differentiable manifold with some Riemmanian metric $g$, so that $(M,g)$ has a nice volume form and one can define $L^2(M,g)$ as the completion of $C^\infty(M)$ under ...
3
votes
1answer
113 views

Cardinality of the set of Boolean subalgbras of the lattice of projections on a Hilbert space.

A simple question I've managed to gey myself quite confused about. Given a Hilbert space H, what do we know about the cardinality of (a) the set P(H) of projection operators onto H (equivalently, ...
0
votes
1answer
169 views

Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?

Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain. How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous? I define $H^{\frac 1 ...
0
votes
2answers
71 views

References for well-posedness of weak solutions to Stefan problem

Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)? ...
1
vote
1answer
183 views

Is the space of test functions separable? [closed]

Consider the space $\mathcal D(\mathbb{R}^n)$ of smooth functions (in the sense of having continuous derivatives of all orders) which are compactly supported. Endow it with its usual topology, i.e., ...
0
votes
3answers
187 views

dual space of a subspace of the space of bounded measures

Let $\mathcal{M}=\mathcal{M}(\mathbb{R})$ be the space of bounded measures. Equipped with the weak convergence, the dual space of $\mathcal{M}$ is $\mathcal{C}_b(\mathbb{R})$ consisting of continuous ...
2
votes
0answers
172 views

Why distributions as functionals? [closed]

Why do we generalize functions by functionals on Schwartz Spaces, beyond the fact that it simply works? There should be a deeper reason why Schwartz considered functionals. Excited for answers, Alex.
1
vote
2answers
131 views

How to show this integral on boundary of Lipschitz domain is finite?

Sorry for asking a basic question but this did not get answered on M.SE. Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that $$\int_{\partial\Omega} ...
2
votes
0answers
89 views

metric entropy for Lipschitz functions

Suppose $(X,d)$ is a metric space of unit diameer and let $F$ be the collection of all $1$-Lipschitz functions mapping $X$ to $[-1,1]$, equipped with the sup-norm $||\cdot||_\infty$. I am interested ...
12
votes
2answers
279 views

compact-open topology on $B(H)$

In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces. Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
0
votes
2answers
116 views

Defining surface integral on boundary of $C^1$-domain

Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to ...
1
vote
1answer
157 views

A linear algebraic q-difference equation [SOLVED]

I would like to solve the following algebraic linear q-difference equation: \begin{equation} a\left(x\right)f\left(x\right)=f\left(qx\right) \end{equation} The parameter $q$ is real, positive and ...
2
votes
0answers
77 views

Constant in Maximal sobolev regularity

We know the following evolution equation \begin{equation} \left\{ \begin{array}{llc} v_t=A v+f,\\ v(0)=0. \end{array} \right. \end{equation} $A$ generates a bounded analytic semigroup on a Banach ...
2
votes
0answers
59 views

System of 2 linear q-difference equations with singular matrix

I would like to solve the following algebraic linear system of q-difference functional equations: \begin{cases} ...
3
votes
2answers
243 views

Dirichlet energy of a harmonic function bounded above by the energy of the boundary function?

$\Omega$ is a domain in $R^2$ with sufficient smooth boundary. Given an absolutely continuous function f difined on $S^1$($[0,2\pi]$). Then there exists an unique harmonic function u defined on ...