Levinson’s operator inequality
Jadranka Mićić
University of Zagreb, Faculty of Mechanical Engineering and Naval Architecture, Ivana
Lučića 5, 10000 Zagreb, Croatia
[email protected]
Joint work with Josip Pečarić and Marjan Praljak
In 1964, Norman Levinson proved that the inequality
n
X
n
n
n
X
X
X
pi f (xi ) − f (
p i xi ) ≤
pi f (yi ) − f (
pi yi )
i=1
i=1
i=1
i=1
holds, for every f : (0, 2c) → R, f 000 ≥ 0 and pi , xi , yi , i = 1, 2, . . . , n, such that pi > 0,
P
n
i=1 pi = 1, 0 ≤ xi ≤ c and x1 + y1 = x2 + y2 = . . . = xn + yn = 2c.
We consider the assumptions on the function f such that Levinson’s inequality holds for
Hilbert space operators.
We give our main result:
Let (X1 , . . . , Xn ) be an n−tuple and (Y1 , . . . , Yk ) be a k−tuple of self-adjoint operators
Xi , Yj ∈ Bh (H) with spectra contained in [mx , Mx ] and [my , My ], respectively, such that mx <
Mx ≤ c ≤ my < My . Let (Φ1 , . . . , Φn ) be an n−tuple and (Ψ1 , . . . , Ψk ) be a k−tuple of positive
Pn
Pk
linear mappings Φi , Ψj : B(H) → B(K), such that i=1 Φi (1H ) = 1K and i=1 Ψi (1H ) = 1K .
Let f ∈ C([mx , My ]) and c ∈ (mx , My ) such that there exists a constant A for which the function
F (x) = f (x) − A2 x2 is operator concave on [mx , c] and operator convex on [c, My ]. If
#
#
" n
" k
n
k
2
2
X
X
X
A
A X
Φi (Xi )
≤ C2 :=
Ψi (Yi )
C1 :=
Φi Xi2 −
Ψi Yi2 −
2 i=1
2
i=1
i=1
i=1
then
n
X
i=1
n
k
k
X
X
X
Φi f (Xi ) − f
Φi (Xi ) ≤ C1 ≤ C2 ≤
Ψi f (Yi ) − f
Ψi (Yi ) .
i=1
i=1
(1)
i=1
Further, using conditions on the spectra of operators we obtain that Levinson’s operator
inequality (1) holds without operator concavity and operator convexity of f .
We also study order among quasi-arithmetic means under the same conditions.