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      Questions tagged [fa.functional-analysis]

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

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      3
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      1answer
      79 views

      Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds

      On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
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      0answers
      22 views

      Domain of definition of a hamiltonian with delta(contact) potential

      I am having a hard time making sense of the so-called "delta function potential well" in quantum theory. The Hamiltonian operator is defined as (with $\mathscr D_H\subset \mathscr H=L^2(\mathbb R)$) $$...
      3
      votes
      1answer
      107 views

      $L^2$ bound and Sobolev spaces

      Let $f \in L^2(\mathbb R)$ be a function such that $$\vert f \vert_{\alpha}:=\sup_{h>0}h^{-\alpha}\Vert f(\bullet+h)-f \Vert_{L^2}< \infty$$ for some $\alpha \in (0,1).$ I would like to know ...
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      vote
      0answers
      48 views

      Are norm one projections in uniformly convex spaces unique?

      Let $X$ be a uniformly convex Banach space and let $M\subset X$ be a $1$-complemented subspace. Is there a unique norm-one projection onto $M$ or might there be more than one? In case the answer is ...
      2
      votes
      2answers
      121 views

      Reference request on Min-Max theorem

      Consider the following min-max problem $$\inf_{x\in M} \sup_{y\in N} F(x,y),$$ where $F: M\times N\to\mathbb R$ is Lipschitz and $y\mapsto F(x,y)$ is concave for all $x\in M$. Could we derive $\...
      4
      votes
      0answers
      90 views

      $L^2$ bound and interpolation of Hölder norm

      Consider the function $$F(x):=\int_{\mathbb R} f(t+x)f(t-x) \ dt .$$ Clearly, we have by Cauchy-Schwarz $$\vert F(x) \vert\le \Vert f \Vert^2_{L^2} $$ $$\vert F'(x)\vert\le 2\Vert f' \Vert_{L^2} \...
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      0answers
      99 views

      What is the structure of a Banach space $X$ when $Y$ and $X/Y$ are hereditarily indecomposable?

      Assume that $X$ is a separable Banach space and $Y$ a closed subspace such that $Y$ and $X/Y$ are hereditarily indecomposable (HI). The general question is what is the possible structure of $X$. ...
      3
      votes
      2answers
      150 views

      Graph with complex eigenvalues

      The question I am wondering about is: Can the discrete Laplacian have complex eigenvalues on a graph? Clearly, there are two cases where it is obvious that this is impossible. 1.) The graph is ...
      2
      votes
      1answer
      63 views

      Properties of the topology of sequential convergence $\tau_{seq}$

      Let $(X,\tau)$ be a Hausdorff space. Denote by $\tau_{seq}$ the topology on $X$ whose closed sets are the sequentially $\tau$-closed subsets of $X$. I have read that $\tau_{seq}$ has the following ...
      -1
      votes
      1answer
      52 views

      Is this kind of interpolation correct?

      Let $f=\sum f_j$ be a finite sum. Assume that $$ \|f\|_2\le(\sum\|f_j\|_2^2)^\frac12$$ $$\|f\|_\infty\le C\max_j\|f_j\|_\infty$$ Then can we conclude that for $2<p<\infty$ $$\|f\|_p\le C^{1-\...
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      0answers
      25 views

      Numerical range of the first derivative operator on $\{ u \in H^1(0,1): u(1)=0 \}$

      I need to calculate the numerical range of the operator $T:D(T)\subseteq L^2(0,1) \to L^2(0,1)$ defined by $$D(T):=\{ u \in H^1(0,1): u(1)=0 \}, \ Tu:=u', \ u \in D(T),$$ where $H^1(0,1)$ is the ...
      2
      votes
      1answer
      77 views

      Regarding outer function being the quotient of two outer functions

      Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
      2
      votes
      1answer
      125 views

      Is there (fast) fourier transform for vector convolution?

      Given a list of variables $u_1,\dots,u_m\in\mathbb R$ and $v_1,\dots,v_n\in\mathbb R$ the standard convolution is defined $$U*V(t)={\sum_{i}} u_iv_{t-i}.$$ Given a list of vectors $u_1,\dots,u_m\in\...
      1
      vote
      1answer
      95 views

      Significance and motivation for outer functions

      Let $\mathbb{D}$ and $\mathbb{T}$ denote the open unit disk and unit circle in $\mathbb{C}$ respectively. We write $Hol(\mathbb{D})$ for the space of all holomorphic functions on $\mathbb{D}.$ The ...
      1
      vote
      0answers
      36 views

      Existence and uniqueness for semilinear problem

      Consider the following problem: $$-\Delta u + [(u)^+]^\alpha = 0,$$ where $(\cdot)^+$ is the positive part function and $\alpha >0$. How does the theory of monotone operators provide existence ...

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