We establish the orthogonality of the range and the kernel of a normal derivation with respect to the unitarily invariant norms associated with norm ideals of operators. Related orthogonality results for certain nonnormal derivations are also given.

Let (H) denote the algebra of operators on a Hilbert space H into itself. Let d= δ or , where δAB : (H)→ (H) is the generalized derivation δAB(S)=AS−SB and AB : (H) → (H) is the elementary operator AB(S) = ASB−S. Given A,B,S ∈ (H), we say that the pair (A,B) has the property PF(d(S)) if dAB(S) = 0 implies dA∗B∗(S) = 0. This paper characterizes operators A,B, and S for which the pair (A,B) has p...

Kernel methods perform nonlinear learning in high-dimensional reproducing kernel Hilbert spaces (RKHSs). Even though their large model-capacity leads to high representational power, it also incurs substantial risk of overfitting. To alleviate this problem, we propose a new regularization approach, nearorthogonality regularization, which encourages the RKHS functions to be close to being orthogo...