Trace Inequalities of Lipschitz Type for Power Series of Operators on Hilbert Spaces
Keywords:
Banach algebras of operators on Hilbert spaces, Power series, Lipschitz type inequalities, Jensen’s type inequalities, Trace of operators, Hilbert-Schmidt normAbstract
Let f (z) = ∑αn zn be a function defined by power series with complex coefficients and convergent on the open disk D(0, R) ⊂ C, R > 0. We show, amongst other that, if T, V ∈ B1 (H), the Banach space of all trace operators on H, are such that ∥T ∥1 , ∥V ∥1 < R, then f (V ), f (T ), f ′((1 − t)T + tV) ∈ B1 (H) for any t ∈ [0, 1] and
tr [f (V )] − tr [f (T )] = ∫ tr [(V − T )f’ (1 − t)T + tV] dt.
Several trace inequalities are established. Applications for some elementary functions of interest are also given.
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