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Tensorial and Hadamard product inequalities of Schwarz type for selfadjoint operators in Hilbert spaces
Get PDFLet H be a Hilbert space. In this paper we show among others that,
if the functions f, g : I ⊂ R →[0,∞) are continuous and A, B are selfadjoint
operators with spectra Sp (A) , Sp (B) ⊂ I, then
f2 (A) ⊗ g2 (B) + g2 (A) ⊗ f2 (B)
≥
h
f2(1−λ) (A) g2λ (A)
i
⊗
h
f2λ (B) g2(1−λ) (B)
i
+
h
f2λ (A) g2(1−λ) (A)
i
⊗
h
f2(1−λ) (B) g2λ (B)
i
≥ 2 [f (A) g (A)] ⊗ [f (B) g (B)]
for all λ ∈ [0, 1] . We also have the following inequalities for the Hadamard
product
f2 (A) ◦ g2 (B) + g2 (A) ◦ f2 (B)
≥
h
f2(1−λ) (A) g2λ (A)
i
◦
h
f2λ (B) g2(1−λ) (B)
i
+
h
f2λ (A) g2(1−λ) (A)
i
◦
h
f2(1−λ) (B) g2λ (B)
i
≥ 2 [f (A) g (A)] ◦ [f (B) g (B)]
for all λ ∈ [0, 1] .departmental bulletin pape
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