ACTA SCIENTIARUM MATHEMATICARUM最新文献

Let (mathcal {U}) be an algebra with center (mathcal {Z(U)}). A mapping (phi :mathcal {U}rightarrow mathcal {U}) is centralizing if (phi (a)a-aphi (a)in mathcal {Z(U)}) for all (ain mathcal {U}). We prove that any continuous centralizing linear map (phi ) on a proper (H^{*})-algebra (mathcal {U}) with (mathcal {U}=ell ^{2}(Gamma , mathcal {U}_{gamma })) ( each (mathcal {U}_{gamma }) is a minimal closed ideal of (mathcal {U})) is of the form (phi (a)=ca+mu (a)), (ain mathcal {U}), where (cin ell ^{infty }(Gamma )) and (mu :mathcal {U}rightarrow mathcal {Z(U)}) is a continuous linear map. Then we examine the automatic continuity of centralizing linear maps on Banach algebras and by using it, a characterization of proper (H^{*})-algebras based on the automatic continuity of centralizing linear maps is given.

A criterion on the similarity of a (bounded, linear) operator T on a (complex, separable) Hilbert space ({mathcal {H}}) to a contraction of class (C_{cdot 0}) with finite unequal defects is given in terms of shift-type invariant subspaces of T. Namely, T is similar to such a contraction if and only if there exists a finite collection of (closed) invariant subspaces ({mathcal {M}}) of T such that the restriction (T|_{{mathcal {M}}}) of T on (mathcal M) is similar to the simple unilateral shift and the linear span of these subspaces ({mathcal {M}}) is ({mathcal {H}}). A sufficient condition for the similarity of an absolutely continuous polynomially bounded operator T to a contraction of class (C_{cdot 0}) with finite equal defects is given. Namely, T is similar to such a contraction if the (spectral) multiplicity of T is finite and (B(T)=mathbb O), where B is a finite product of Blaschke products with simple zeros satisfying the Carleson interpolating condition (a Carleson–Newman product).

We determine systems of the first order ordinary differential equations such that their group of symmetries contains a three-dimensional Lie subgroup G. We represent the basis vectors of the Lie algebra (mathfrak {g}) of G by vector fields in the three-dimensional real space. Two cases are distinguished according to whether the infinitesimal generators of (mathfrak {g}) do not contain any component or contain component with respect to the independent variable of the system.

This paper is a sequel to Jung (Bull Aust Math Soc 97: 133–140, 2018) that was originally written concurrently with Jung (Bull Aust Math Soc 97: 133–140, 2018). In that paper we transferred the discussions in Androulakis (Int Eq Op Th 65: 473–484, 2009) and Popov (J Funct Anal 265: 257–265, 2013) concerning almost invariant half-spaces for operators on complex Banach spaces to the context of operators on Hilbert space, and we gave slightly simpler proofs of the main results in Androulakis (Int Eq Op Th 65: 473–484, 2009) and Popov (J Funct Anal 265: 257–265, 2013) in that context. In the present paper we discuss a consequence of the main construction in Jung (Bull Aust Math Soc 97: 133–140, 2018) for the restriction to a half-space of a certain large class of operators on Hilbert space.

We obtain various upper bounds for the numerical radius w(T) of a bounded linear operator T defined on a complex Hilbert space (mathcal {H}), by developing the upper bounds for the (alpha )-norm of T, which is defined as (Vert TVert _{alpha }= sup left{ sqrt{alpha |langle Tx,x rangle |^2+ (1-alpha )Vert TxVert ^2 }: xin mathcal {H}, Vert xVert =1 right} ) for ( 0le alpha le 1 ). Further, we prove that

For (0le alpha le 1 le beta ,) the operator T is called ((alpha ,beta ))-normal if (alpha ^2 T^*Tle TT^*le beta ^2 T^*T) holds. Note that every invertible operator is an ((alpha ,beta ))-normal operator for suitable values of (alpha ) and (beta ). Among other lower bounds for the numerical radius of an ((alpha ,beta ))-normal operator T, we show that

where (Re (T)) and (Im (T)) are the real part and imaginary part of T, respectively.

Algorithms for the computation of the (weighted) geometric mean G of two positive definite matrices are described and discussed. For large and sparse matrices the problem of computing the product (y=Gb), and of solving the linear system (Gx=b), without forming G, is addressed. An analysis of the conditioning is provided. Substantial numerical experimentation is carried out to test and compare the performances of these algorithms in terms of CPU time, numerical stability, and number of iterative steps.

Let (mathscr {L}({mathscr {H}})) be the algebra of all bounded linear operators acting on an infinite-dimensional complex Hilbert space ({mathscr {H}}), and denote by (gamma (T)) the reduced minimum modulus of any operator (Tin mathscr {L}({mathscr {H}})). We obtain the form of all bijective linear maps (Phi ) on (mathscr {L}({mathscr {H}})) for which (gamma (Phi (T))=gamma (Phi (S))) whenever (T,~Sin mathscr {L}({mathscr {H}})) are two operators equivalent by unitaries. We also obtain similar results when the reduced minimum modulus is replaced by the minimum modulus or the surjectivity modulus.

The purpose of this article is to prove fixed point theorems for new classes of acyclic mappings known as almost acyclic contractions and utmost acyclic contractions in the framework of a uniformly convex Banach space. Further, it is interesting to observe that a best proximity point theorem for cyclic contractions is elicited as an application of one of the fixed point theorems.

The geometry of the Birkhoff polytope, i.e., the compact convex set of all (n times n) doubly stochastic matrices, has been an active subject of research. While its faces, edges and facets as well as its volume have been intensely studied, other geometric characteristics such as the center and radius were left off, despite their natural uses in some areas of mathematics. In this paper, we completely characterize the Chebyshev center and the Chebyshev radius of the Birkhoff polytope associated with the metrics induced by the operator (ell ^p_n)-norms for the range (1 le p le infty ).

In the first of this series of two articles, we studied some geometrical aspects of the Birkhoff polytope, the compact convex set of all (n times n) doubly stochastic matrices, namely the Chebyshev center, and the Chebyshev radius of the Birkhoff polytope associated with metrics induced by the operator norms from (ell _n^p) to (ell _n^p) for (1 le p le infty ). In the present paper, we take another look at those very questions, but for a different family of matrix norms, namely the Schatten p-norms, for (1 le p < infty ). While studying these properties, the intrinsic connection to the minimal trace, which naturally appears in the assignment problem, is also established.