BANACH JOURNAL OF MATHEMATICAL ANALYSIS
Year:2008 | Volume:2 | Issue:2|Papers Count:16

Let Aj (j = 1, 2, . . . , n) be strict contractions on a Hilbert space. We study an n × n operator-matrix:Hn(A1,A2, . . . ,An) = [(I- Aj*Ai)-1]ni,j=1.For the case n = 2, Hua [Inequalities involving determinants, Acta Math. Sinica, 5 (1955), 463–470 (in Chinese)] proved positivity, i.e., positive semidefiniteness of H2(A1,A2). This is, however, not always true for n = 3. First we generalize a known condition which guarantees positivity of Hn. Our main result is that positively of Hn is preserved under the operator Möbius map of the open unit disc D of strict contractions.

For analytic functions f in the open unit disc U, a generalization operator Dlf(z) of Sălăgean operator is introduced. Some properties for Dlf(z) are discussed in the present paper.

We shall give a norm inequality equivalent to the grand Furuta inequality, and moreover show its reverse as follows: Let A and B be positiveoperators such that 0 < m £B £ M for some scalars 0 < m < M andh := M/m > 1. Then||A1/2 {A_t/2 (A -r/2B (r-t){(p-t)a+r/1-t+rAr/2}1/a A-t/2}1/PA1/2 || £ K(hr-t , (p-t) s+r/1-t+r) 1/ps || A 1-t+r/2 B r-t A 1-t+r/2 || (p-t)a+r/pa (1-t+r) for 0 £ t £  1, p ³ 1, s ³1 and r ³ t ³0, where K(h, p) is the generalized Kantorovich constant. As applications, we consider reverses related to the Ando-Hiai inequality.

Remarks about strengthening of the triangle inequality and its reverse inequality in normed spaces for two and more elements are collected. There is also a discussion on Fischer–Muszély equality for n-elements in a normed space. Some other estimates which follow from the triangle inequality are also presented.

We survey some old and new results concerning weighted norm inequalities of sum and product form and apply the theory to obtain limit point conditions for second order differential operators of Sturm-Liouville form defined in Lp spaces. We also extend results of Anderson and Hinton by giving necessary and sufficient criteria that perturbations of such operators be relatively bounded. Our work is in part a generalization of the classical Hilbert space theory of Sturm-Liouville operators to a Banach space setting.

The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an operatorvalued inner product, we discuss operator versions for the above inequalities and give simple conditions that the equalities hold.

An eigenvalue problem is considered where the eigenvalue appears in the domain and on the boundary. This eigenvalue problem has a spectrum of discrete positive and negative eigenvalues. The smallest positive and the largest negative eigenvalue l±1 can be characterized by a variational principle. We are mainly interested in obtaining non trivial upper bounds for l-1. We prove some domain monotonicity for certain special shapes using a kind of maximum principle derived by C. Bandle, J.v. Bellow and W. Reichel in [J. Eur. Math. Soc., 10 (2007), 73–104]. We then apply these bounds to the trace inequality.

In this paper, necessary and sufficient conditions for the validity of the Hardy inequality for the case q < 0, p > 0 and for the case q > 0, p < 0 are derived.

For 0 < r < 1 and 1£ p£q < ¥ we find necessary and sufficient conditions for the validity of the following inequality: (òba u(x) (òxa | g(x)-g (t)| r w(t)dt)q/r dx)1/q £ c(òba v(x)| g’ (x)|p dx)1/pwhere u(·), v(·), and w(·) are weight functions.

Hardy–Sobolev–type inequalities associated with the operator L := x .Ñ are established, using an improvement to the Sobolev embedding theorem obtained by M. Ledoux. The analysis involves the determination of the operator semigroup {e_tL*L} t>0.

A characterization of nonlinear spectral radius preserving maps is obtained for the usual and triple products of nonnegative matrices.

Let W be a compact convex subset of Rd and let (Ln)nÎN be a sequence of positive linear operators that map C(W) into itself. We establish two Korovkin-type theorems in which the limit of the sequence of operators is not necessarily the identity.

Some new refined multidimensional Hardy-type inequalities for p³2 and their duals are derived and discussed. Moreover, these inequalities hold in the reversed direction when 1 < p £ 2. The results obtained are based mainly on some new results for superquadratic and subquadratic functions. In particular, our results further extend the recent results in [J.A. Oguntuase and L.-E. Persson, Refinement of Hardy’s inequalities via superquadratic and subquadratic functions, J. Math. Anal. Appl., 339 (2008), no. 2, 1305– 1312] to a multidimensional setting.

In this paper, we show reverses of the Golden–Thompson type inequalities due to Ando, Hiai and Petz: Let H and K be Hermitian matrices such that mI £ H, K£MI for some scalars m£M, and let a Î [0, 1]. Then for every unitarily invarint norm |||e(1−a)H+aK||| £S(ep(M−m)) 1/p||| (epH #a pK)1/p|||holds for all p > 0 and the right-hand side converges to the left-hand side as p ¯ 0, where S(a) is the Specht ratio and the a-geometric mean X #a Y is defined as X #a Y= X1/2 (X-1/2 YX-1/2) X1/2        for all 0£a£1 for positive definite X and Y.

Mixed-mean inequalities for integral power means over centered and uncentered spheres are proved. There from we deduce the Hardy type inequalities for corresponding averaging operators. Moreover, we discuss estimates related to the spherical maximal functions. 

While Professor J.E. Pečarić and the interviewer are meeting each other in three conferences in Iran, Hungary and Croatia, they had some conversations concerning life, ideas and contributions of Professor Pečarić to Mathematics. This paper presents them.