SOME OPERATOR BOUNDS EMPLOYING COMPLEX INTERPOLATION REVISITED
SOME OPERATOR BOUNDS EMPLOYING COMPLEX INTERPOLATION REVISITED
复制标题
重新审视使用复杂插值的一些算子界限
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Y. Tomilov
中科院分区：
文献类型：
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作者：
F. Gesztesy;Y. Latushkin;F. Sukochev;Y. Tomilov
我们重温并推广了这类算子值函数的已知界
$$T^{-Z}_{1}ST^{-1+z}_{2}，；zinar{Sigma}；=；left{zinmathbb{C}|mathrm{Re}{z}in[0，1]
夜}$$
在关于线性算子S和Tj，j=1，2的各种假设下，我们特别指出了一些可分复Hilbert空间(Mathcal{H}_{j}，j；=；1，2，)中的自伴算子和扇形算子Tj的情况，并假定S(分别为S*)是稠定闭算子映射don((S)的子集mathcal{H}_{1}；mathm{into}；mathcal{H}_{2}；(mathm{resp.}，mathm{don}(S^{*}))的子集Mathm{into}；数学{H}_{1})，相对于(T_{1}；(mathm{res.，}；T^{*}_{2})相对有界)。利用复插值方法，S的广义极分解和Loewner-Heinz不等式的(变体)，我们建立的界将导致下列类型的不等式：给定的(k；in；(0，inty))，
$$|overline{T^{-Z}_{1}ST^{-1+z}_{2}}|_{mathcal{B}(mathcal{H}_{1}，mathcal{H}_{2})}；leq；N_{1}N{2}e^{k(mathrm{Im}(z))^2+k；MATHROM{Re}(Z)[1-MATHROM{Re}(Z)]+(4k)^{-1}(HETA_{1}+HETA_{2})^2}imes|ST^{-1}_{1}|^{1-{mathrm{Re}}(z)}_{mathcal{B}(mathcal{H_{1}，Mathcal{H}_{2}})}|S^{*}(T^{*}_{2})^{-1}|^{{mathrm{Re}}(z)}_{mathcal{B}(mathcal{H_{2}，数学{H}_{1}})，qquad Zinar{ar{Sigma}}$$
这也意味着，
$$|overline{T^{-Z}_{1}ST^{-1+z}_{2}}|_{mathcal{B}(mathcal{H}_{1}，数学{H}_{2})}；方程；N_{1}N{2}e^{(HETA_{1}+HETA_{2})[x(1-x)]^{1/2}}imes|ST^{-1}_{1}|^{1-{x}_{mathcal{B}(mathcal{H_{1}，mathcal{H}_{2}})}|S^{*}(T^{*}_{2})^{-1}|^{x}_{mathcal{B}(mathcal{H_{2}，数学{H}_{1}})，Qquad Xin[0，1]}$$
设Tj有有界虚幂，即对某(N_(J)geqslant；1mathm{and}；heta；geqslant；0)，
$$|T^{is}_{j}|_mathcal{B(H)}；leq；N_{j}e^{heta_{j}|S|}，四个sinmathbb{R}，j；=；1，2$$
。我们还用迹理想(数学{B}(数学{H}_{1}，数学{H}_{2}))，(数学{B}_{p}(数学{H}_{1}，数学{H}_{2})，；p；in；[1，inty))得到类似的界。所采用的方法是初级的，主要依赖于Hadamard的三线定理和Loewner-Heinz不等式。
We revisit and extend known bounds on operator-valued functions of the type
$$T^{-Z}_{1}ST^{-1+z}_{2},;zinar{Sigma};=;left{zinmathbb{C}|mathrm{Re}{z}in[0,1]
ight}$$
under various hypotheses on the linear operators S and Tj , j = 1, 2. We particularly single out the case of self-adjoint and sectorial operators Tj in some separable complex Hilbert space (mathcal{H}_{j},j;=;1,2,) and suppose that S (resp., S*) is a densely defined closed operator mapping dom ((S)subseteq mathcal{H}_{1}; mathrm{into};mathcal{H}_{2};(mathrm{resp.}, mathrm{dom}(S^{*})subseteq mathcal{H}_{2};mathrm{into};mathcal{H}_{1}), relatively bounded with respect to (T_{1};(mathrm{resp.,};T^{*}_{2})). Using complex interpolation methods, a generalized polar decomposition for S, and (a variant of) the Loewner–Heinz inequality, the bounds we establish lead to inequalities of the following type: Given (k;in;(0,infty)),
$$|overline{T^{-Z}_{1}ST^{-1+z}_{2}}|_{mathcal{B}(mathcal{H}_{1},mathcal{H}_{2})};leq;N_{1}N{2}e^{k(mathrm{Im}(z))^2+k; mathrm{Re}(z)[1-mathrm{Re}(z)]+(4k)^{-1}( heta_{1}+ heta_{2})^2} imes|ST^{-1}_{1}|^{1-{mathrm{Re}}(z)}_{mathcal{B}(mathcal{H_{1},mathcal{H}_{2}})}|S^{*}(T^{*}_{2})^{-1}|^{{mathrm{Re}}(z)}_{mathcal{B}(mathcal{H_{2},mathcal{H}_{1}})},qquad zinar{ar{Sigma}}$$
which also implies,
$$|overline{T^{-Z}_{1}ST^{-1+z}_{2}}|_{mathcal{B}(mathcal{H}_{1},mathcal{H}_{2})};leq;N_{1}N{2}e^{( heta_{1}+ heta_{2})[x(1-x)]^{1/2}} imes|ST^{-1}_{1}|^{1-{x}_{mathcal{B}(mathcal{H_{1},mathcal{H}_{2}})}|S^{*}(T^{*}_{2})^{-1}|^{x}_{mathcal{B}(mathcal{H_{2},mathcal{H}_{1}})},qquad xin[0,1]}$$
assuming that T j have bounded imaginary powers, that is, for some (N_{j}geqslant;1mathrm{and}; heta;geqslant;0),
$$|T^{is}_{j}|_mathcal{B(H)};leq;N_{j}e^{ heta_{j}|s|},quad sinmathbb{R}, j;=;1,2$$
. We also derive analogous bounds with (mathcal{B}(mathcal{H}_{1},mathcal{H}_{2})) replaced by trace ideals, (mathcal{B}_{p}(mathcal{H}_{1},mathcal{H}_{2}),;p;in;[1,infty)). The methods employed are elementary, predominantly relying on Hadamard’s three-lines theorem and the Loewner–Heinz inequality.
影响因子：
0.8
作者：
Batty C
通讯作者：
Batty C