All Questions
11 questions
2
votes
1
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136
views
Is there a scalar product which makes orthonormal the family of complex functions $ (f_n)_{ n \geq 1 } $?
Let $ (f_n)_{ n \geq 1 } $ be a family of complex functions defined as follow,
$ \forall n \geq 1 $,
$$ f_n (z) = \dfrac{1}{n^{z}} $$
I would like to ask you if it is possible to construct a ( non-...
- 595
4
votes
1
answer
196
views
A kind of holomorphicity of maps on Hilbert space
Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?:
1)For every open set $U\subset H$ and every ...
- 356
2
votes
0
answers
55
views
A holomorphic map into a Hilbert space with prescribed orthogonality
This is a variation of my previous question.
Let $X\subset \mathbb{C}^n$ be a domain, and let $L:X\times X\to \mathbb{C}$ be such that $L(x,x)>0$, $L(y,x)=\overline{L(x,y)}$ and $L(\cdot,y)$ is ...
- 5,529
8
votes
0
answers
251
views
Struggling with a proof in the preprint 'Hermitian geometry on resolvent set'
I have been struggling for awhile with a particular argument in the paper below. I posted the question first on MathSE, but I got no answers. I understand however that MO might be an overreach for ...
- 81
3
votes
2
answers
470
views
inner product on matrix spaces of multivariate polynomials?
Let $H_{n,d}=\mathbb{R}_d[x_1,..,x_n]$ be the space of $n$-variate homogeneous degree $d$ polynomials, $D=D^\top\in \mathbb{N}^{m\times m}$ a symmetric $m\times m$ matrix. Consider the space $P_D$ of ...
- 14k
2
votes
1
answer
113
views
Norm of vector-valued holomorphic functions
Let $G$ be a connected simply connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space.
Q1. Which functions $F:G\to(0,+\infty)$ are such that there is a holomorphic $f:G\to H\backslash ...
- 5,529
6
votes
1
answer
319
views
Derivatives of norm of vector-valued holomorphic functions
Let $G$ be a connected domain in $\mathbb{C}^{n}$, let $H$ be a Hilbert space and let $f,g:G\to H\backslash \{0\}$ be holomorphic (in my particular situation they are also injective, but I don't think ...
- 5,529
6
votes
2
answers
671
views
Holomorphy of a function with values in a Hilbert space
Denote by $\mathbb C^\infty $ the Hilbert space $\ell^2 (\mathbb C)$. Fix $1\leq N,M \leq \infty$, and let $U$ be an open subset of $\mathbb C^N $. Following Mujica's book "complex analysis in Banach ...
- 395
1
vote
0
answers
109
views
Zeros of functions constituting a Riesz-basis for the Paley-Wiener space
I have a short question which first requires some slightly elaborate definitions:
Let $(e_n)$ be a Riesz-basis for a Hilbert space $\mathcal{H}$ with biorthogonal basis $(g_n)$, i.e. $\langle e_m, ...
- 11
4
votes
0
answers
112
views
status of Invariant subspace problem on Krein Space
What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.
- 2,106
0
votes
1
answer
2k
views
What is the orthonormal basis for the Bergman space on the disk?
[EDIT by YC: the original question's title asked about a basis for the Hardy space on the disk. It is clear from the actual question that what was meant was the Bergman space.]
In arXiv:0310.5297, ...
- 22.8k