Abstract. We give an inequality relating the operator norm of T and the numerical radii of T and its Aluthge transform. It is a more precise estimate of the numerical radius than Kittaneh's result [Studia Math. 158 (2003)]. Then we obtain an equivalent condition for the numerical radius to be equal to half the operator norm.

We have previously demonstrated that in mice, the decrease in serum thyroxine (T(4)) level by polychlorinated biphenyls (PCBs) occurs without an increase in the UDP-glucuronosyltransferase (T(4)-UDP-GT) for T(4) glucuronidation, although the PCB-induced decrease in rats is generally thought to occur through induction of T(4)-UDP-GT, UGT1A1, and UGT1A6. In the present study, to further clarify the relationship between the decrease in serum T(4) level and the increase in UGT1A activity by PCB in rats, we examined the relationship using Wistar rats and Gunn rats, a mutant strain of Wistar rats deficient in UGT1A isoforms. The serum total T(4) level was markedly decreased not only in the Wistar rats but also in the Gunn rats 4 days after treatment with a PCB, Kanechlor-500 (KC500, 100 mg/kg) or 2,2',4,5,5'-pentachlorobiphenyl (PentaCB, 112 mg/kg), and there was no significant difference in magnitude of the decrease between the two rat strains. At the same time, the level and activity of T(4)-UDP-GT were significantly increased by treatment with either KC500 or PentaCB in Wistar rats but not in Gunn rats. In addition, no significant change in the level of serum total triiodothyronine (T(3)) and thyroid-stimulating hormone by the KC500 treatment was observed in either Wistar or Gunn rats. Furthermore, significant decrease in the activity of hepatic type-I deiodinase, which mediates the deiodization of T(4) and T(3), by treatment with KC500 or PentaCB was observed in both Wistar and Gunn rats. From the serum of KC500- or PentaCB-treated Wistar and Gunn rats, mono- and di-hydroxylated PCB metabolites, which would bind to T(4) binding serum protein (transthyretin), were detected. In conclusion, the present results suggest that the decrease in serum total T(4) level by either KC500 or PentaCB in Gunn rats was not dependent on the increase in hepatic T(4)-UDP-GT activity. The findings further suggest that the PCB-mediated decrease in serum T(4) level might occur, at least in part, through formation of the hydroxylated PCB metabolites. Furthermore, even in Wistar rats, the PCB-mediated decrease in serum T(4) level might occur not only through the increase in hepatic T(4)-UDP-GT but also via formation of hydroxylated PCB metabolites.

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Abstract. The Riemannian mean on the convex cone of positive definite matrices is a kind of geometric mean of n -matrices which is an extension of the geometric mean of two-matrices. In this paper, we derive the Ando-Hiai inequality for the Riemannian mean which is an extension of the well-known Ando-Hiai inequality of two-matrices. Moreover, we shall show an extension of a characterization of chaotic order. Lastly, we will give a negative answer for the problem whether the same results are satisfied or not for other geometric means of n -matrices.Mathematics subject classification (2010): Primary 47A64; Secondary 47A63, 47L25.

Abstract. We show that if T = H + iK is the Cartesian decomposition of T ∈ B(H ), then for α, β ∈ R, sup α 2 +β 2 =1 αH + βK = w(T ). We then apply it to prove that if A, B, X ∈ B(H ) and 0 ≤ mI ≤ X, thenwhere Re(T ) denotes the real part of an operator T . A refinement of the triangle inequality is also shown.