Functons of perturbed pairs of noncommuting dissipative operator

A. B. Aleksandrov ; V. V. Peller

Algebra i analiz, Tome 34 (2022) no. 3, pp. 93-114

Résumé

Let $f$ be a function belonging to the nonhomogeneous analytic Besov space $(\mathrm{\text{�}}_{\infty,1}^1)_+(\mathbb{R}^2)$. For a pair $(L,M)$ of not necessarily commuting maximal dissipative operators, the function $f(L,M)$ is introduced as a densely defined linear. For $p\in[1,2]$, we prove that if $(L_1,M_1)$ and $(L_2,M_2)$ are pairs of not necessarily commuting maximal dissipative operators such that the two difeerences $L_1-L_2$ � $M_1-M_2$ belong to the Schatten–von Neumann class $\mathbf{S}_p$, then for every $f$ in $(\mathrm{\text{�}}_{\infty,1}^1)_+(\mathbb{R}^2)$ the operator difference $f(L_1,M_1)-f(L_2,M_2)$ belongs to $\mathbf{S}_p$ and the following Lipschitz-type estimate holds true: $ \|f(L_1,M_1)-f(L_2,M_2)\|_{\mathbf{S}_p} \le\mathrm{const}\,\|f\|_{\mathrm{\text{�}}_{\infty,1}^1}\max\big\{\|L_1-L_2\|_{\mathbf{S}_p},\|M_1-M_2\|_{\mathbf{S}_p}\big\}. $

Détail

Keywords: dissipative operator, Haagerup tensor product, Haagerup-type tensor products, semispectral measure, functions of noncommuting operators, Lipschitz-type estimates for functions of operators, Schatten–von Neumann classes.
Mots-clés : Besov classes

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