Let $\mathcal{H}$ be a separable Hilbert space with the unit operator I, let A be a bounded linear operator in $\mathcal{H}$ with a Schatten–von Neumann Hermitian component $(A-A^{\ast })/2i$ ( $A^{\ast }$ means the operator adjoint to A) and let $f(z)$ be a function analytic on the spectra of A and $A^{\ast }$. For $f(A)$ we obtain the representation in the form of the sum of a normal operator and a quasi‐nilpotent Schatten–von Neumann operator $V{}_f$, and estimate the norm of $V{}_f$. That estimate gives us an inequality for the norm of the resolvent ${(\lambda I-f(A))}^{-1}$ of $f(A)$ $(\lambda \in \mathbf{C})$. Applications of the obtained estimate for ${(\lambda I-f(A))}^{-1}$ to operator equations, whose coefficients are operator functions, and to perturbations of spectra are also discussed.

中文翻译：

---

具有Schatten–von Neumann Hermitian分量的算子函数的三角表示

让 $\mathcal{H}$是单位算符I的可分离希尔伯特空间，令A是其中的有界线性算子$\mathcal{H}$ 带有Schatten–von Neumann Hermitian分量 $（\mathrm{一种}-{\mathrm{一种}}^{\ast }）/2\mathrm{一世}$ （ ${\mathrm{一种}}^{\ast }$表示运算符与A伴随）$F（ž）$是分析A和A光谱的函数${\mathrm{一种}}^{\ast }$。对于$F（\mathrm{一种}）$ 我们以正态算子和拟幂零的Schatten–von Neumann算子之和的形式获得表示形式 $V{}_F$，并估算 $V{}_F$。这个估计使我们对解析者的规范不平等${（\lambda \mathrm{一世}-F（\mathrm{一种}））}^{-\mathrm{1个}}$ 的 $F（\mathrm{一种}）$ $（\lambda \in \mathbf{C}）$。所获得的估计数用于${（\lambda \mathrm{一世}-F（\mathrm{一种}））}^{-\mathrm{1个}}$ 还讨论了系数为算子函数的算子方程以及频谱的扰动。