On jointly concavity of some trace functions
Bekjan T.N. Ospanov K.N.
1 May 2023Elsevier Inc.
Linear Algebra and Its Applications
2023#664147 - 164 pp.
Let M be a finite von Neumann algebra with a normal faithful finite trace τ, L0(M) be the set of all measurable operators with respect to (M,τ) and μt(x) be the generalized singular number of x∈L0(M). Set L0(M)+={x:x∈L0(M),x≥0} and M++={x:x∈M,x≥0andxis invertible}. We prove that if f:[0,∞)→[0,∞) is an operator concave function, 0j is a continuous positive linear map from L0(Mj) to L0(M) with Φj(Mj)⊂M, where Mj is finite von Neumann algebra, j=1,2,⋯,n, then for 0≤t<τ(1) ∫tτ(1)μv((∑j=1nΦj(f(xjp)))s)dvand∫tτ(1)μv((∑j=1nΦj(f(xj)p))s)dv are jointly concave in (x1,x2,⋯,xn)∈L0(M1)+×L0(M2)+×⋯×L0(Mn)+. We also prove that if f:(0,∞)→(0,∞) is an operator concave function, Φj is a strictly positive linear map from finite von Neumann algebra Mj to M, j=1,2,⋯,n, 0v((∑j=1nΦj(f(xj−p)))−s)dvand∫tτ(1)μv((∑j=1nΦj(f(xj)−p))−s)dv are jointly concave in (x1,x2,⋯,xn)∈M1++×M2++×⋯×Mn++.
Finite von Neumann algebra , Positive operator , Submajorization
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Astana IT University, Astana, 010000, Kazakhstan
Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana, 010008, Kazakhstan
Astana IT University
Faculty of Mechanics and Mathematics
10 лет помогаем публиковать статьи Международный издатель
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