Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
3,436 questions with no upvoted or accepted answers
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Variational Problems with Subsidiary Conditions
I am studying from
Gelfand, I.M.; Fomin, S.V., Calculus of variations. Transl. from the Russian and edited by Richard A. Silverman., Mineola, NY: Dover Publications. vii, 232 p. (2000). ZBL0964....
- 11
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114
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Outer product $\sum_i |k_{x_{i}}(\cdot)\rangle\langle k_{x_{i}}(\cdot)|$ of reproducing kernel functions as identity operator in RKHS?
In a separable Hilbert space $\mathcal{H}$, given a complete orthonormal basis $\{|e_i\rangle\}$, the identity operator can be written as $\mathbb{1} = \sum_i |e_i\rangle\langle e_i|$. Now if this ...
- 11
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45
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Is research of the Hyers-Ulam stability of this functional equation already conducted?
The functional equation in question is of the type $f(g(x))=g(f(x))$, where $f$ is the unknown function. Are there existing research already conducted on the Hyers-Ulam stability of this generalized ...
- 21
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215
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Minimal projections and pure states in scattered $C^*$ algebras
The question I have came up when reading a paper of Ghasemi and Koszmider https://arxiv.org/abs/1611.00221 on scattered $C^*$ algebras and the non-commutative Cantor-Bendixson derivatives. Throughout ...
- 56
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336
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Existence of solution for Poisson equation in Markov chain
Consider $X_n\in \mathcal{X}$ a controlled Markov chain taking value in a compact set $\mathcal{X}$ with action $a\in \mathcal{A}$, where the action set $|\mathcal{A}|$ is finite.
(In particular, we ...
- 185
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110
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Are almost positive functionals close to positive functionals?
This is a bit of an open-ended question... Let $S$ be an operator algebra (or an operator system) and consider a functional $\nu:M\to \mathbb{C}$
that satisfies
$$\vert \nu(a)\vert \ge -\varepsilon \...
- 19
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105
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Generalize characterization of upper semicontinous functions
Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$
It is easy to show that the following two statements are equivalent:
$(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
- 623
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202
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Space of analytic function and sequence space $l^p$
Let $\mathbb{D} = \{z:|z|<1\}$ be open unit disc in complex plane. Define space of analytic function:
$N^p=\{f:\mathbb{D} \to \mathbb{C} | f(z)=\sum_{n=0}^{\infty} a_n z^n, \sum_{n=0}^{\infty}|a_n|...
- 259
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183
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Trace theorem for boundary value problem
Consider the inhomogeneous boundary value problem on the infinite strip $(x,y)\in \mathbb{R}\times [0,1]$ defined by
$$\begin{cases}\partial_{x}u + \partial_{y}v=f & {(x,y)\in \mathbb{R}\times (0,...
- 873
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87
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Linear evolution equation $u'(t)=A(t,\omega)u(t)$ with time-dependent random operator
I have had some previous knowledge on evolution equations in a Banach space of the form $$u'(t)=Au(t),$$ where $A$ generates some strongly continuous operator semigroup. Now I am looking at a problem ...
- 151
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180
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Implicit function theorem for operators
Let $P: (-a,a) \rightarrow \Psi_h^0(\mathbb{R}),$ be a pseudodifferential operator in Weyl quantization with $(-a,a) \ni z \mapsto P(z)$ depending smoothly on this parameter $z$. Note that this ...
- 115
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304
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$C^*$-algebras with non-trivial center
The center $Z(A)$ of an algebra $A$ is the set of all those elements that commute with all other elements. If $A$ is the algebra of compact operators on a Hilbert space $H$, then $A^{**}$ is the ...
- 4,461
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124
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Singular value decomposition in two spaces (reference in Russian paper?)
Let $H$ be a Hilbert space and $X$ be a Banach space such that $H \cap X$ is dense in both.
Now, let $T$ be an operator such that $T: H \rightarrow H$ and $T:X \rightarrow X$ exists in the sense that ...
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151
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Continuity of the spectrum under weaker notions of convergence
Let $T:X\to X$ be a linear operator on a Banach space $X$.
We know that the spectrum of $T$ is an upper semicontinuous
function of $T$ for the uniform convergence: that is, if $T_n:X\to X$ is a ...
- 757
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94
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Riesz transform on almost periodic functions?
It is well established that the Riesz transform is well-defined for $f\in L^p(\mathbb{R}^d)$ via$$
\mathcal{R}_jf(x) = c_d\lim_{\epsilon\to 0}\int_{|x-y|>\epsilon}\frac{(x^j-y^j)f(y)}{|x-y|^{d+1}}\,...
- 663
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96
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Infimum of equivalent measures
Suppose I have a functional of the form
$$
F(\mathbb{P})\triangleq \int_{\mathbb{R}^d} \int_{\Omega}f(x,\omega)\mathbb{P}(d\omega)m(dx),
$$
where $m$ is the Lebesgue measure and $\mathbb{P}$ is a ...
- 173
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148
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Fourier–Stieltjes as the dual space of the full group algebra
I know that this fact is classical, but I can't find the proof of it. How to proof that $B(G)=(C^*(G))^*$? As I understood, I can take a functional $F: \ell_1(G) \to \mathbb{C}$, and there is one-to-...
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48
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Existence of arbitrarily small invariant neighborhood
Let $A$ be a bounded linear operator from a Banach space $M$ to itself.
Suppose that $\rho(A)<1$ where $\rho(A)$ is the spectral radius of $A$.
For any $\varepsilon>0$, does there exist an open ...
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80
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What is the character space of $\mathcal P(K)$?
Let $K$ be a compact subset of $\Bbb C$. Let $\mathcal P(K)$ be the closed algebra generated by the complex polynomials on $K$.
What is the character space $\Phi_{\mathcal P(K)}$ of $\mathcal P(K)$?...
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154
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Lyapunov stability for nonlinear PDEs
Where can I find a theorem about Lyapunov stability for the equation in Hilbert space?
$$
y' = Fy,
$$
where $F : V \to V'$ is a nonlinear operator , $y' \in L^2(0,T,V')$, $V$ is a Hilbert space.
...
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220
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About the projection on the unit sphere
Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
- 5,529
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109
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Two tensor product norms inducing different topologies on the space of simple tensors
Are there two Normed spaces $V,W$ for which the algebraic tensor product $V\otimes W$ admits two different norms, both satisfying $\parallel x \otimes y \parallel= \parallel x \parallel. \...
- 356
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50
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Verifying general assumption for parabolic PDE
I've got some problems verifying an assumption for a parabolic PDE. Namely, let $(V,H,V^*)$ be a Gelfand-Triple, $u_0 \in V$, $\psi\colon V \to \mathbb{R}$ convex and lower-semicontinuous and $a\colon ...
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84
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Existence of a minimizer of a functional involving a power q
I asked this question on math.stackexchange (https://math.stackexchange.com/questions/2052565/existence-of-a-minimizer-of-a-functional-involving-a-power-q) but did not get any answers so I am trying ...
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66
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How to define spectral multiplier for −Δ?
Put $e_{n}(t)=e^{int}=\prod_{j=1}^{d}e^{in_{j}t_{j}}$, $t\in \mathbb T^d, n\in \mathbb Z^d.$ ($t=(t_{1},..., t_d), n=(n_1,..., n_d)$)
We note that $\{e_{n}\}_{n\in \mathbb Z^d}$ forms an orthonormal ...
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66
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Characterization of the weak completion of $L^2(\mathbb{R}^d)$
The completion $\overline{L^2_w(\mathbb{R}^d)}$ of $L^2_w(\mathbb{R}^d)$ (i.e. the completion of $L^2(\mathbb{R}^d)$ endowed with the $\sigma(L^2(\mathbb{R}^d),L^2(\mathbb{R}^d))$ topology) is ...
- 488
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105
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compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
- 1,385
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194
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Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
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331
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Verifying a claim regarding $H^1$ weak convergence and $L^2$ strong convergence on a surface
I'm reading a paper whose first section discussed $H^1$ maps defined on star-shaped sets, but I got stuck in verifying a claim for quite a while. I'm now thinking the claim is wrong, but it's hard to ...
- 1,350
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128
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determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
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154
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Kasparov's descent homomorphism for higher KK groups
I am currently trying to understand equivariant $KK$-theory. I think I roughly get the idea of Kasparov's descent homomorphism
$$KK^G(A,B) \rightarrow KK(A \rtimes G,B \rtimes G).$$
but what still ...
- 19
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123
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Generalization of concave envelope
Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
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What is the analogue of expansive matrix for automorphisms?
We say an invertible $n \times n$ matrix with entries in $\Bbb R^n$ is expansive if the absolute values of all of its eigenvalues exceed $1$. An easy calculation also shows that if we consider a ball ...
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Reference for a text book on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$
We need basic reference on the Toeplitz operators $T:l^{\infty}(\mathbb Z, \mathbb R^2)→l^{\infty}(\mathbb Z, \mathbb R^2)$. Usually text books cover much more subtle case of $T:l^{\infty}(\mathbb Z_+,...
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94
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Space spanned by pointwise squares of basis functions
Consider the Hilbert space $L^2(\Omega)$ over some Euclidean domain $\Omega$.
Let $F=\{f_i;i\in\mathbb N\}$ be an orthonormal basis of this space consisting of functions in $L^2(\Omega)\cap L^4(\Omega)...
- 8,086
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205
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Essential self-adjointness of tensor product of operators
Operators $A_i, i=1,2,\ldots,5$ are considered on Hilbert spaces $\mathcal{H}_i$. Operator $A_i$ is essentially self-adjoint on $D^{ess}(A_i) \subset \mathcal{H}_i$.
Consider operator
\begin{equation}...
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256
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Significance of Tikhonov matrix
I am looking for a tutorial on Tikhonov matrix, in the sense what it can do or it cannot do. The definition of the matrix can be obtained in the wikipedia link. https://en.wikipedia.org/wiki/...
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136
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heat kernel for powers of some degenerate elliptic operators
Let $\Omega$ be a bounded open domain in $R^{n}$ with smooth boundary and $X=(X_{1},X_{2},\cdots,X_{m})$ be a system of real smooth vector fields defined on $\Omega\subset \mathbb{R}^{n}$. If $X$ ...
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90
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Monotone operator subgradient
I am trying to solve a PDE of the form $\mathcal{A}u'(t) + \partial\Psi[u(t)] \ni 0$ where $\mathcal{A}$ is a skew-symmetric, linear, monotone operator, $\Psi$ is convex, and $\partial \Psi$ is the ...
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238
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Continuation of strictly monotone function in $R^n$
Let $f(x_1,...,x_n)$ be $C^0$ continuous function $R^n\to R$ defined on a compact domain $A\subset R^n$. Let $f$ be strictly monotonously increasing w.r.t. every argument in the domain of definition. ...
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83
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What is optimal distance between inverse of convolution operator?
I am looking for a measure to find the optimal distance measure between inverse of an convolution operator $A$ and say another convolution operator $B$. I want my measure to be sharp that mean when $B$...
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86
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Least restrictive condition such that $-y''(x)+q(x)y(x)=\lambda y(x)$ has two solutions
Let $\lambda \in \mathbb{R}$ and consider some function $q$ on the interval $[0,1].$
What is known to be the least restrictive condition on $q$ such that there are two linearly independent $H^2$ ...
- 11
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135
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infinite dimensional funtional ito calculus
I've been reading into functional Ito calculus and everything I've come across deals with processes generated by finite dimensional semimartingales. In Dupire's 2009 landmark paper he speaks about ...
- 5,405
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664
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$W^{2,p}$ regularity of elliptic PDEs with Neumann boundary condition
Given an elliptic PDE with Neumann boundary condition
\begin{align}
\left\{
\begin{aligned}
-\sum_{i,j=1}^N\partial_i(a_{ij}\partial_j u)+cu&=f &&\mbox{in}\,\,\,\Omega, \\
\sum_{i,j=1}^Na_{...
- 393
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0
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94
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About a particular definition of a Hessian of a function of tuples of matrices
Say I have a function $L : (W_1,..,W_{H+1}) \rightarrow \mathbb{R}$ i.e it takes a tuple of $n$ matrices of different dimensions and computes a number from them.
Then I see being defined a ...
- 2,246
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117
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The eigenfunction of modified $1$-laplace equation?
Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
- 679
1
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0
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94
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$L^\infty$ bounds for pseudo-differential equations of parabolic type
It is well-known that if the solution of $u_t=u_{xx}$, with $t>0$ and $x\in\mathbb R$, is bounded, then $a(t)=\sup_{x\in \mathbb R}u(x,t)$ is non-increasing, while
$b(t)=\inf_{x\in \mathbb R}u(x,t)$...
- 2,933
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148
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References for the Sturm oscillation theorem
What is the most general form of the Sturm oscillation theorem?
So far I have only seen cases that treat either unbounded intervals or weighted $L^2$ spaces. I would be especially interested in ...
- 263
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158
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Domain of operator
Let be $\lambda\in C^{*}$. Consider the following operator:
$ T_{\lambda}=-\Delta_{R^{2}}++\frac{\lambda^{2} }{4} (x^{2}+y^{2})+i\lambda N$,
where
$N=(x \frac{d }{dy} -y \frac{d }{dx})$ ,
...
- 135
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0
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116
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Projective tensor product continuous?
For $V$, $W$ Banach spaces, is the canonical morphism $B(V) \times_\pi B(W) \to B(V \hat{\otimes}_\pi W)$ continuous, where $B(V)$ denotes the set of all linear bounded endomorphisms with operator ...
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