Anti-N-Order Polynomial Daugavet Property on Banach Spaces

Paper presented at the 5th Strathmore International Mathematics Conference (SIMC 2019), 12 - 16 August 2019, Strathmore University, Nairobi, KenyaWe generalize the notion of the anti-Daugavet property (a-DP) to the anti-N-order Almasi room, polynomial Daugavet property (a-NPDP) for Banach spaces. The characterization SBS of the a-NPDP is through the spectral information; however, it is well-known in nonlinear theory that there is no suitable notion of the spectra for nonlinear operators resulting into enormous structural challenges to the known characterization techniques for the a-DP. To bypass some of the problems, we establish that a good spectrum of a nonlinear operator is one whose associated eigenvectors are of unit norm and study the a-NPDP for locally uniformly convex or smooth Banach spaces (luacs); in particular, we prove that locally convex or smooth finite dimensional Banach spaces have the a-mDP for rank-I polynomials and then extend this result to innite dimensional luacs Banach spaces. Besides, we prove that locally uniformly convex Banach spaces have the a-NPDP for compact polynomials if and only if their norms are eigenvalues, and moreover, uniformly convex Banach spaces have the a-NPDP for continuous polynomials if and only if their norms belong to the approximate spectra. As a consequence of these results, we conclude that all continuous In-homogeneous polynomials that satisfy the N-order polynomial Daugavet equation on a uniformly convex Banach space such as Lr-spaces for 1 < r < 1 and Hilbert spaces have nontrivial invariant subspaces; this result was not known.Mbarara University of Science and Technology, Ugand

Polynomially compact bilinear endorphisms of finite type of Banach algebras

Paper presented at the 4th Strathmore International Mathematics Conference (SIMC 2017), 19 - 23 June 2017, Strathmore University, Nairobi, Kenya.Kamowitz’s classical result on a compact endomorphism T of a commutative Ba- nach algebra A asserts that ∩T ∗n(Xr ) is finite where T ∗ is the adjoint of T and Xr),is the set of multiplicative linear functionals on A. This paper extends the underlying Kamowitz’s result to absolutely r-summing operators for 1 ™ r &lt; ∞ or more generally polynomially compact endomorphisms as well as bilinear operators of finite type generated by Polynomially compact operators of a commutative Banach algebra. Keywords: Polynomial compactness; Endomorphism; Algebra; absolute summability; bilinear operators.Mbarara University of Science and Technology P. O. Box 1410 Mbarara – Ugand

On the Banach algebra numerical range

Paper presented at the 4th Strathmore International Mathematics Conference (SIMC 2017), 19 - 23 June 2017, Strathmore University, Nairobi, Kenya.Let A denote a unital Banach algebra and SA denote its unit sphere. It was proved by F. F. Bonsalland J. Duncan that when the Banach algebra is unital, the algebra numerical rangeis identical to a subset of itself. Basing on this fact, we have established that the union overall elements of SA of the sets of support functionals for the unit ball at each x ∈ SA i.e.Sx∈SD(A, x) is equal to the set of normalized states i.e. D(A, eˆwhen the Banach algebraAˆ).ˆeeeeA is unital (eˆ is the unit element). The implication is that even when the algebra is smooth, theset of normalized states i.e. D(A, e) is not a singleton. Hence algebra numerical range is notsingleton except when the element is a scalar multiple of the unit. Also, consider the statementsP1 and P2;(P1) The union over all elements of SA of the sets of support functionals for the unit ball ateach x ∈ SA i.e. Sx∈S D(A, x) is equal to the set of normalized states i.e. D(A, eˆ-component algebra numerical(P2) The algebra numerical range i.e. V (A, a) is equal to the eˆrange i.e. V (A, a, e).We have proved that the statements P1 and P2 are equivalent under a suitable condition. Further,F.F Bonsall and J. Duncan proved that for a unital Banach algebra, the numerical radius is anequivalent algebra norm. In that proof, the inequalities k &gt; v(a) and k ≥ 1 &quot;a&quot; were used toconclude that 1 &quot;a&quot; ≤ v(a) (e is the irrational number 2.718...). However, these two inequalities lead to the undesired inequality 1 &quot;a&quot; ≥ v(a). In this dissertation we improve on this proof bymerging the property of the complex roots of unity used by Bonsall and Duncan together withthe geometric series argument and making a different choice of the arbitrary element b ∈ A ofnorm less than one to derive the desired inequality 1 &quot;a&quot; ≤ v(a).andˆD(A, eˆe) = 1}.) = {f ∈ SAt : f (ˆ Elements of D(A, x) are called support functionals for the unit ball at x ∈ SA while elements of D(A, e) are called normalized states. For each a ∈ A and x ∈ SA defineV (A, a, x) = {f (ax) : f ∈ D(A, x)}Mbarara University of Science and Technology, Mbarara, Uganda Maseno University, Maseno, kenya - [email protected] Mbarara University of Science and Technology, Mbarara, Ugand

An invariant subspace problem for multilinear operators on finite dimensional spaces

We introduce the notion of invariant subspaces for multilinear operators from which the invariant subspace problems for multilinear and polynomial operators arise. We prove that polynomial operators acting in a finite dimensional complex space and even polynomial operators acting in a finite dimensional real space have eigenvalues. These results enable us to prove that polynomial and multilinear operators acting in a finite dimensional complex space, even polynomial and even multilinear operators acting in a finite dimensional real space have nontrivial invariant subspaces. Furthermore, we prove that odd polynomial operators acting in a finite dimensional real space either have eigenvalues or are homotopic to scalar operators; we then use this result to prove that odd polynomial and odd multilinear operators acting in a finite dimensional real space may or may not have invariant subspaces