Second-order trace formulas

Pub Date : 2024-03-22 DOI:10.1002/mana.202200295
Arup Chattopadhyay, Soma Das, Chandan Pradhan
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引用次数: 0

Abstract

Koplienko [Sib. Mat. Zh. 25 (1984), 62–71; English transl. in Siberian Math. J. 25 (1984), 735–743] found a trace formula for perturbations of self-adjoint operators by operators of Hilbert–Schmidt class B 2 ( H ) $\mathcal {B}_2(\mathcal {H})$ . Later, Neidhardt introduced a similar formula in the case of pairs of unitaries ( U , U 0 ) $(U,U_0)$ via multiplicative path in [Math. Nachr. 138 (1988), 7–25]. In 2012, Potapov and Sukochev [Comm. Math. Phys. 309 (2012), no. 3, 693–702] obtained a trace formula like the Koplienko trace formula for pairs of contractions by answering an open question posed by Gesztesy, Pushnitski, and Simon [Zh. Mat. Fiz. Anal. Geom. 4 (2008), no. 1, 63–107, 202; Open Question 11.2]. In this paper, we supply a new proof of the Koplienko trace formula in the case of pairs of contractions ( T , T 0 ) $(T,T_0)$ , where the initial operator T 0 $T_0$ is normal, via linear path by reducing the problem to a finite-dimensional one as in the proof of Krein's trace formula by Voiculescu [Oper. Theory Adv. Appl. 24 (1987) 329–332] and Sinha and Mohapatra [Proc. Indian Acad. Sci. Math. Sci. 104 (1994), no. 4, 819–853] and [Integral Equations Operator Theory 24 (1996), no. 3, 285–297]. Consequently, we obtain the Koplienko trace formula for a class of pairs of contractions using the Schäffer matrix unitary dilation. Moreover, we also obtain the Koplienko trace formula for a pair of self-adjoint operators and maximal dissipative operators using the Cayley transform. At the end, we extend the Koplienko–Neidhardt trace formula for a class of pairs of contractions ( T , T 0 ) $(T,T_0)$ via multiplicative path using the finite-dimensional approximation method.

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二阶轨迹公式
科普连科 [Sib.Mat.25 (1984), 62-71; English transl.25 (1984), 62-71; English transl. in Siberian Math.25 (1984), 735-743] 发现了希尔伯特-施密特类算子对自相关算子扰动的迹公式。后来,奈德哈特在[数学通报 138 (1988),7-25]中通过乘法路径引入了一对单元的类似公式。2012 年,波塔波夫和苏科切夫[Comm. Math. Phys. 309 (2012),no. 3, 693-702]通过回答格兹特西、普什尼茨基和西蒙[Zh. Mat. Fiz. Anal. Geom. 4 (2008),no. 1, 63-107, 202; Open Question 11.2]提出的一个开放问题,得到了类似科普连科迹式的成对收缩迹式。在本文中,我们通过线性路径,将问题简化为有限维问题,提供了科普连科迹线公式在成对收缩情况下的新证明,其中初始算子是正常的,正如沃伊库勒斯库对克雷恩迹线公式的证明[Oper.24 (1987) 329-332] 以及 Sinha 和 Mohapatra [Proc. Indian Acad. Sci.因此,我们利用 Schäffer 矩阵单元扩张,得到了一类成对收缩的科普连科迹线公式。此外,我们还利用 Cayley 变换得到了一对自相关算子和最大耗散算子的科普连科迹线公式。最后,我们利用有限维近似法,通过乘法路径扩展了一类成对收缩的科普连科-奈德哈特迹公式。
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