Abstract
The Berezin transform (A ) over tilde of an operator A, acting on the reproducing kernel Hilbert space H = H(Omega) over some (non-empty) set Omega, is defined by (A) over tilde(lambda) = (A (k) over cap (lambda), (k) over cap (lambda) ) (lambda is an element of Omega), where (k) over cap lambda = k(lambda)/parallel to k(lambda)parallel to is the nor-malized reproducing kernel of H. The Berezin number of an operator A is defined by ber(A) = sup(lambda is an element of Omega)vertical bar(A) over tilde(lambda)vertical bar = sup(lambda is an element of Omega)vertical bar < A (k) over cap (lambda),(k) over cap (lambda >)vertical bar. In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space H, then
ber(AX +/- XA) <= ber(1/2) (A(*)A + AA(*)) ber(1/2) ((XX)-X-* + XX*)
and
ber(2) (A(*)XB) <= parallel to X parallel to(2) ber(A(*)A)ber(B-* B).
We also prove the multiplicative inequality
ber(AB) <= ber(A)ber(B).