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Results tagged with hilbert-spaces
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user 153260
A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.
0
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1
answer
107
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A property of the canonical dual frame in a Hilbert space
Let $\{ g_n \} $ be a frame in a separable Hilbert space $H$. Then the frame operator $S:H\to H$ defined as
\begin{equation} S f := \sum_{n=1}^\infty (f,g_n)g_n \end{equation}
is a Hilbert space isomo …
1
vote
1
answer
43
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Wold decomposition of toral endomorphisms
Suppose that $A\in M_d(\mathbb{Z})$ is a $d \times d$ matrix with non zero determinant and suppose that $\mathbb{T}^d$ is the $d$-dimensional torus. Then one can define an operator on $L^2(\mathbb{T}^ …
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$L^2$ space of Hilbert-Schmidt operator valued functions
No, I don't think so. For an explicit example, for $v,u\in L^2(\mathbb{R})$, let me use the notation $v\otimes w$ for the rank $1$ operator $(v\otimes w) (f) = (w,f) v$. Then consider any fixed $g\in …
1
vote
Accepted
Orthonormal bases in RKHSs via interpolating sequences
It is true that if a sequence $(k_n)$ is interpolating for $M(\mathcal{H})$, then the normalized Kernel vectors $g_n:=K_{k_n}/\Vert K_{k_n} \Vert_\mathcal{H} $ form a Riesz system in $\mathcal{H}$. O …