Questions tagged [fa.functional-analysis]

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

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34 views

Bounded approximation property and affine subspaces of codimension $1$

Let $E$ be a Banach space with the bounded approximation property and let $M\subseteq E$ be an affine subset of co-dimension $1$, and let $C\subseteq M$ be a closed and convex subset of $M$ with non-...
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25 views

Pertrubing the approximation property from the Lipschitz-free space to stay in the Wasserstein space

Let $(X,d,x)$ be a separable pointed metric space and let $\mathcal{F}(X)$ be its Arens-Eells (also called its Lipschitz-Free space; in the case where $X$ is Banach) space. We view the $1$-...
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22 views

Knowledge on weighted integral operators?

There are tons of books and a huge literature on the properties of the following integral operator: \begin{equation} T(f) = \int_{\mathcal{X}} K(x,\cdot)f(x)dx, \end{equation} where $K(x,z)$ is, say, ...
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46 views

product of two generalized functions

Let $f_n$ and $g_n$ two generalized functions such that : the product $f_n g_n$ is well defined for all $n\in \mathbb{N}$, which means $WF(f_n)+WF(g_n)$ does not contain an element of the form $(x,0)$...
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52 views

Differential properties of the Radon transform

I am looking for a reference for the following fact: Let $f \in L_1(\mathbb{R}^{d})$ that has compact support and that is also in $C^{s}(\mathbb{R}^{d})$. We define the Radon transform of $f$ as: \...
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33 views

Compact embedding of Lipschitz continuous functions

Let $(X,d,\mu)$ be a metric measure space, not necessarily with $\mu(X)<\infty$. I would like to study the embedding of $W^{1,2}(X)\cap \mathrm{Lip}(X)$ into $L^2(X)$. Are there simple conditions ...
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1answer
21 views

Weak majorizations for sum of two hermitian matrices

Let $A$ and $B$ be two $n\times n$ hermitian matrices. Does $U^{*}AU+B \prec_{w} A+B$ for any unitary matrix $U$? Here the notation $``\prec_{w}"$ stands for the weak majorization, that is, $x\prec_{...
4
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1answer
139 views

The largest $\ell_p$-norm of a sum of rows of a Sylvester-Hadamard-Walsh matrix

Given any $n\in\mathbb N$, consider the the Sylvester-Hadamard-Walsh matrix $M=(a_{i,j})_{i,j\in 2^n}$ of size $2^n\times 2^n$ and for a number $p\in[1,\infty)$, let $$\nu_{n,p}=\max_{F\subseteq 2^n}\...
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38 views

Are feature maps into RKHSs are embeddings?

Let $H$ be a Hilbert space, $X$ be a metric space, and $F: X\rightarrow H$ be a continuous function. Define the RKHS associated to $F$ as: $$ H_{F} = \{ g: X \to \mathbb{R} | \exists w \in H, g(x) = \...
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1answer
76 views

Prove $\int_\Omega \left(\rho_{1} \ln \frac{\rho_{1}}{\rho_{2}}\right)dx dy \leq C\int_\Omega |\rho_1-\rho_2|dxdy$ for $0 \le \rho_1, \rho_2 \in L^1$

Let $\rho_1, \rho_2 \in L^1(\Omega;\mathbb R_+)$ such that $\int \rho_i|\ln \rho_i| < \infty$. Is it true that there exists a constant $C>0$ such that \begin{align*} \int_\Omega \left(\rho_{1} \...
4
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1answer
125 views

Separable subalgebras of non-separable reflexive Banach algebras

Let $A$ be a non-separable reflexive Banach algebra. Every separable subspace of $A$ is contained in a separable 1-complemented subspace [Lindenstrauss,1966]. It is straightforward to show that every ...
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55 views

Every Borel linearly independent set has Borel linear hull (reference?)

I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone. Theorem. The linear hull of any linearly independent Borel set in a Polish ...
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44 views

Small bi-Lipschitz maps

Let $d,n$ be positive integers where $\mathbb{R}^d$ is viewed as a subspace of $\mathbb{R}^n$. What is the smallest $M>0$ for which there exists some Lipschitz map $L:\mathbb{R}^n\rightarrow \...
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59 views

Pointwise convergence of kernels of Hilbert-Schmidt operators

Lately I was discussing different types of convergence for Hilbert-Schmidt operators and during that discussion we ended up talking about pointwise convergence of Fourier series. I have already asked ...
4
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1answer
235 views

Taylor coefficients of Hadamard product

I imagine this to be a very classical question in complex analysis: Consider the Hadamard product $$g(\mu) = \prod_{n=1}^{\infty}E_1(\mu z_n),$$ where $E_1(z):=(1-z)e^z$ is the first elementary ...
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62 views

A convex version of the small uncountable cardinal $\mathfrak b$

Let us recall that $\mathfrak b$ is the smallest cardinality of a subset of $\omega^\omega$, which cannot be covered by countably many compact subsets of $\omega^\omega$. The definition of $\mathfrak ...
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84 views

Does a holomorphic function with logarithmic growth at the boundary have $L^2$ boundary values?

Let $f(z)$ be a holomorphic function on the unit disc, with logarithmic growth at the boundary: $$ |f(z)| = \mathcal O\bigg(\log\Big(\frac{1}{1-|z|}\Big)\bigg). $$ Does it follow that the (...
3
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1answer
109 views

Spectrum Cauchy-Euler operator

A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations We consider the operator $$(Lf)(x) = \...
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105 views
+50

Tensor product of representations on a compact quantum group

Let $\mathbb{G}$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz) with function algebra $(C(\mathbb{G}), \Delta)$. Let $X \in M(B_0(H)\otimes C(\mathbb{G}))$ and $Y \in M(B_0(K)\...
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1answer
282 views

Estimating the growth of the Taylor coefficients given the growth of the function at the boundary

Let $f(z)=\sum a_nz^n$ be a Taylor series that converges for $|z|<1$ and satisfies $$ |f(z)|\le \frac{1}{(1-|z|)^{k}} $$ for some fixed $k>0$. Question: What can I deduce about the growth of the ...
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2answers
179 views

Minimal injective extension is rigid

Let $V$ be an operator system. Definition 1: A pair $(W, \kappa)$ is called extension of $V$ if $W$ is an operator system and $\kappa: V \to W$ is a unital complete isometry. Definition 2: An ...
4
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1answer
154 views

If a completely positive unital map admits a completely positive unital left inverse, it is a complete isometry

Let $T$ be an injective operator system and $U$ be an arbitrary operator system. Let $\varphi: T \to U$ be a unital completely positive map and $\psi: U \to T$ be a unital completely positive map with ...
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51 views

Regular inclusions: $\{b\in B:E(b^*b)=0\}$ is a two-sided ideal

From [Donsing-Pitts-2008, theorem 4.8]: For $A\subseteq B$ a regular inclusion, with $A$ abelian, and $E:B\to A$ its unique conditional expectation it holds: The left ideal $$L(E):=\{b\in B:E(b^*b)=0\...
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2answers
256 views

Implicit function theorem with continuous dependence on parameter

Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$. Let $f:X\times P\to Y$ be a continuous map such that for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...
3
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0answers
86 views

Approximation of a linear functional by linear continuous functionals

Let $X$ be a locally convex space, $T$ a balanced convex compact set in $X$, and $f:X\to\mathbb{C}$ a linear functional which is (not necessarily continuous on $X$, but) continuous on $T$. It is not ...
4
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1answer
88 views

Is the "hereditarily indecomposable" property separably determined?

Is it true that a Banach space $X$ is hereditarily indecomposable if every separable closed subspace of $X$ is hereditarily indecomposable?
3
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71 views

Does convolution by a Schwartz function preserve symbol classes?

I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
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0answers
31 views

Systematic approach to Weierstrass factorization

If you want to calculate the Taylor expansion of a function, you only need to know the derivatives of the function at the point of expansion. Is there a similar algorithmic approach that can be ...
5
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0answers
131 views

Absolute summability of multiplication operators on $\ell_p$

A linear bounded operator $T:X\to Y$ between Banach spaces is called absolutely summing if for every unconditionally convergent series $\sum_{i\in\omega}x_i$ in $X$ the series $\sum_{i\in\omega}\|T(...
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39 views

Unbounded subsets of $\ell^+_\infty$ that have non-empty interior and non-trivial unique minimizers

Let $\ell^+_\infty$ be the space of non-negative bounded sequences equipped with the $\sup$ norm topology. Note that the dual of $\ell_\infty$ is the space of finitely additive signed measures on $2^\...
3
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0answers
135 views

What do people call functionals on holomorphic functions and on polynomials?

There are four most important functional spaces in analysis: the space $\mathcal{C}(M)$ of continuous functions on a topological space, the space $\mathcal{E}(M)$ of smooth functions on a smooth ...
4
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1answer
156 views

Example: traceless C*-algebra universally generated by projections

Are there examples of a non-zero C*-algebra which is universally generated by finitely many projections (not all commuting) together with a unit and plus necessarily satisfying some additional ...
2
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1answer
94 views

Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy"

(This question is related to Splitting a space into positive and negative parts but different.) Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative ...
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0answers
514 views

Green's function of Cauchy-Riemann operator on Torus

Consider the torus $\mathbb T^2:=\mathbb C/(\mathbb Z+i \mathbb Z)$ and the operator $$ T = (2D_{\bar z}-\lambda)^{-1} $$ on the torus with periodic boundary conditions. This one is well-defined for $\...
4
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1answer
91 views

The symmetric group of positive integers acting on $\ell^\infty(\mathbb{N})^*$

Let $\mathfrak{S}_\mathbb{N}$ be the symmetric group of all positive integers. Let $\ell^\infty(\mathbb{N})^*$ be the dual space of $\ell^\infty(\mathbb{N})$ equipped with weak*-topology. There is a ...
2
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1answer
119 views

$(\iota \otimes f)(X) = 0$ for all $f \in B^*$ implies $X=0$

Let $A$ and $B$ be $C^*$-algebras. Given $f \in B^*$, we can form the right slice map $$\iota \otimes f: A \otimes B \to A: a \otimes b \mapsto af(b)$$ which extends uniquely to a bounded linear map $$...
3
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1answer
138 views

Discrete singular integrals

Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties: $\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
3
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1answer
112 views

Is the ball ratio theorem for Radon–Nikodým derivative known for general metric spaces?

Given two non-negative Borel measures $\mu$, $\nu$ on $\mathbb{R}^n$, that are finite on compact sets, such that $\nu\ll\mu$, it is well known that $$\frac{d\nu}{d\mu}(x)= \lim_{\epsilon\to 0} \frac{\...
11
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1answer
197 views

(Sharp) Bounds on $E(XYZ)$ given all the bivariate marginals

Suppose $X,Y,Z$ are all real-valued random variables. Suppose I know the joint marginal distributions of $(X,Y)$, $(Y,Z)$ and $(X,Z)$. I want to find bounds on $E(XYZ)$. In the case of bounding $E(XY)$...
2
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1answer
52 views

Elliptic equation on square with periodic boundary values for the solution and it's partial derivatives

Suppose that $\Omega=[0,1]^2$. I will say that a real valued function $u$ on $\Omega$ satisfies periodic boundary values if $$u(x,0)=u(x,1), \;u(0,y)=u(1,y),\;\;\;\text{ for all }x,y\in[0,1].$$ Now ...
0
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2answers
122 views

Example of a linear operator whose graph is not closed

I want an example of a linear operator $T:X\to Y$ such that graph of $T$ is not closed. My thoughts: $T$ must be unbounded. Again by closed graph theorem any unbounded linear map from a Banach space $...
7
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1answer
295 views

Bounding the decrease after applying a contraction operator $n$ vs $n+1$ times

Can we upper bound the convergence rate of $$\max_{\textbf{v}: \left\Vert \textbf{v}\right\Vert_2=1} \left\{ \left\Vert \textbf{T}^n \textbf{v}\right\Vert^2_2 - \left\Vert \textbf{T}^{n+1} \textbf{v}\...
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0answers
68 views

Generator problem for reduced group C*-algebra

(Not sure if it is appropriate or not, if no I will delete the post) Recently I am concerned about the number of generator of $C^{*}_{r}(\mathbb{F}_{k})$, the reduced group algebra of the free group, ...
9
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3answers
529 views

Takesaki theorem 2.6

I originally posted this question on MSE and didn't get a satisfactory answer, even after putting a bounty on it. Hence, I thought I should ask here: Consider the following theorem in Takesaki's book &...
6
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1answer
161 views

Infinite-dimensional projections of linearly independent sets

A subset of a linear space $X$ is called infinite-dimensional if it is not contained in a finite-dimensional linear subspace of $X$. Problem. Let $L$ be an infinite-dimensional subset of the linear ...
8
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0answers
395 views

Why is spectral theory developed for $\mathbb C$

Spectral theory is a fundamental part of operator theory and the spectrum of many operators is investigated throughout the existing literature. And that is for a good reason: If $A$ is some closed ...
-1
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1answer
69 views

Fundamental of a signal

Consider the space $S$ of real functions with the norm $$\|f\|^2 = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} e^{-x^2/2} f^2(x) ~\mathrm{d}x, $$ or any reasonable Euclidean norm such that bounded ...
8
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1answer
246 views

General validity of separation of variables

Let $L$ be any differential operator (not necessarily linear). Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form: Given a boundary ...
5
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1answer
249 views

Making sense of the formula $\operatorname{Det} (I+M )= e^{\operatorname{Tr} \ln (I+M)}$, especially in the infinite dimensional cases

$\DeclareMathOperator\Det{Det}\DeclareMathOperator\Tr{Tr}$In physics literature dealing with quantum field theory, the formula \begin{equation} \Det(I+M) = e^{\Tr \ln(I+M)} \end{equation} appears ...
2
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1answer
154 views

How to choose minimisers in a continuous way

Let $\langle X, X' \rangle$ be a dual pair equipped with the weak and weak* topologies. Let $C$ be a weak* compact subset of $X'$ with nonempty interior. For each $x \in X$, let $M(x)$ be the set of ...

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