# Questions tagged [fa.functional-analysis]

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

7,675
questions

**-1**

votes

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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-...

**1**

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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 $1$-...

**1**

vote

**0**answers

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, ...

**0**

votes

**0**answers

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 $(x,0)$...

**0**

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**0**answers

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:
\...

**0**

votes

**0**answers

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 ...

**1**

vote

**1**answer

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**

votes

**1**answer

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}\...

**0**

votes

**0**answers

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) = \...

**1**

vote

**1**answer

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**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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 \...

**2**

votes

**0**answers

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**

votes

**1**answer

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 ...

**2**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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**

votes

**1**answer

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) = \...

**4**

votes

**0**answers

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)\...

**11**

votes

**1**answer

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 ...

**6**

votes

**2**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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\...

**10**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

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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 ...

**1**

vote

**0**answers

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**

votes

**0**answers

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(...

**0**

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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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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**

votes

**1**answer

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**

votes

**1**answer

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 ...

**13**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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}\...

**0**

votes

**0**answers

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**

votes

**3**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**1**answer

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 ...