Ideal spaces of measurable operators affiliated to a semifinite von Neumann algebra. II

Let \(\tau \) be a faithful semifinite normal trace on a von Neumann algebra \(\mathcal {M}\), let \(S(\mathcal {M}, \tau )\) be the \({}^*\)-algebra of all \(\tau \)-measurable operators. Let \(\mu (t; X)\) be the generalized singular value function of the operator \(X \in S(\mathcal {M}, \tau )\). If \(\mathcal {E}\) is a normed ideal space (NIS) on \((\mathcal {M}, \tau )\), then

$$\begin{aligned} \Vert A\Vert _\mathcal {E}\le \Vert A+\textrm{i} B\Vert _\mathcal {E} \end{aligned}$$

(*)

for all self-adjoint operators \(A, B \in \mathcal {E}\). In particular, if \(A, B \in (L_1+L_{\infty })(\mathcal {M}, \tau )\) are self-adjoint, then we have the (Hardy–Littlewood–Pólya) weak submajorization, \(A \preceq _w A+\textrm{i}B\). Inequality \((*)\) cannot be extended to the Shatten–von Neumann ideals \(\mathfrak {S}_p\), \( 0< p <1\). Hence, the well-known inequality \( \mu (t; A) \le \mu (t; A+\textrm{i} B)\) for all \(t>0\), positive \(A \in S(\mathcal {M}, \tau )\) and self-adjoint \( B \in S(\mathcal {M}, \tau )\) cannot be extended to all self-adjoint operators \(A, B \in S(\mathcal {M}, \tau )\). Consider self-adjoint operators \(X, Y\in S(\mathcal {M}, \tau )\), let K(X) be the Cayley transform of X. Then, \(\mu (t; K(X)-K(Y))\le 2 \mu (t; X-Y)\) for all \(t>0\). If \(\mathcal {E}\) is an F-NIS on \((\mathcal {M}, \tau )\) and \(X-Y\in \mathcal {E}\), then \(K(X)-K(Y)\in \mathcal {E}\) and \(\Vert K(X)-K(Y)\Vert _\mathcal {E}\le 2 \Vert X-Y\Vert _\mathcal {E}\).