On factorization of the shift semigroup
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914296595742720 |
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| author | Bhattacharyya, Tirthankar Rastogi, Shubham Sinha, Kalyan B. U, Vijaya Kumar |
| author_facet | Bhattacharyya, Tirthankar Rastogi, Shubham Sinha, Kalyan B. U, Vijaya Kumar |
| contents | Let $\E$ be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup $§^\E=(S_t^\E)_{t\ge 0}$ on $L^2(\R_+,\E)$ into the product of $n$ commuting contractive semigroups, i.e., characterizes all $n$-tuples of commuting semigroups $(\V_1,\V_2,...,\V_n)$ where $\V_i=(V_{i,t})_{t\ge 0}$ for $i=1,2,...,n$ are semigroups of contractions satisfying $V_{i,t}V_{j,t}=V_{j,t}V_{i,t}$ for all $i$ and $j$ and $S_t^\E=V_{1,t}V_{2,t}\cdots V_{n,t}$ for all $t\ge 0.$ The factorizations are characterized by tuples of self-adjoint operators $\underline{A}=(A_1,A_2,...,A_n)$ and tuples of positive contractions $\underline{B}=(B_1,B_2,...,B_n)$ on $\E$ satisfying certain conditions which are stated in \cref{thm:psi12}. One of the tools of our analysis is a convexity argument using the extreme points of the {\em Herglotz } class of functions
\[P:=\{f:\D\to \C \text{ is analytic}, \Re{f}>0 \text{ and }f(0)=1 \}.\] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_15343 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On factorization of the shift semigroup Bhattacharyya, Tirthankar Rastogi, Shubham Sinha, Kalyan B. U, Vijaya Kumar Functional Analysis Primary: 47D03, 47A65, 47A68 Let $\E$ be a finite dimensional Hilbert space. This note finds all factorizations of the right shift semigroup $§^\E=(S_t^\E)_{t\ge 0}$ on $L^2(\R_+,\E)$ into the product of $n$ commuting contractive semigroups, i.e., characterizes all $n$-tuples of commuting semigroups $(\V_1,\V_2,...,\V_n)$ where $\V_i=(V_{i,t})_{t\ge 0}$ for $i=1,2,...,n$ are semigroups of contractions satisfying $V_{i,t}V_{j,t}=V_{j,t}V_{i,t}$ for all $i$ and $j$ and $S_t^\E=V_{1,t}V_{2,t}\cdots V_{n,t}$ for all $t\ge 0.$ The factorizations are characterized by tuples of self-adjoint operators $\underline{A}=(A_1,A_2,...,A_n)$ and tuples of positive contractions $\underline{B}=(B_1,B_2,...,B_n)$ on $\E$ satisfying certain conditions which are stated in \cref{thm:psi12}. One of the tools of our analysis is a convexity argument using the extreme points of the {\em Herglotz } class of functions \[P:=\{f:\D\to \C \text{ is analytic}, \Re{f}>0 \text{ and }f(0)=1 \}.\] |
| title | On factorization of the shift semigroup |
| topic | Functional Analysis Primary: 47D03, 47A65, 47A68 |
| url | https://arxiv.org/abs/2306.15343 |