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New inequalities of the Kantorovich type for bounded linear operators in Hilbert spaces. (English) Zbl 1142.47007
If the selfadjoint operator A on a Hilbert space H is such that mIAMI, where 0<m<M, then the Kantorovich inequality says that 1Ax,xA -1 x,x(m+M) 2 /(4mM) for any unit vector x in H. In this paper, the author uses Grüss-type inequalities, which he obtained before, and their operator versions to establish inequalities of Kantorovich type for the more general class of operators A satisfying Re[(A * -α ¯I)(βI-A)]0. The Grüss-type inequalities refer to the ones which give upper bounds for |u,v-u,ee,v| for vectors u, v and e in H with e=1 and scalars α, β, γ and δ satisfying u-((α+β)/2)e|β-α|/2 and v-((γ+δ)/2)e|δ-γ|/2. There are also established estimates for A 2 -w(A) 2 and w(A) 2 -w(A 2 ) for the above class of A. Here, w(A) denotes the numerical radius sup{|Ax,x|:xH, x=1} of A.
MSC:
47A12Numerical range and numerical radius of linear operators
47A30Operator norms and inequalities
47A63Operator inequalities
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