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Register a new userTheory of $B(X)$-module -Algebraic module structure of generally-unbounded infinitesimal generators-
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Yoritaka Iwata
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The concept of logarithmic representation of infinitesimal generators is introduced, and it is applied to clarify the algebraic structure of bounded and unbounded infinitesimal generators. In particular, by means of the logarithmic representation, the bounded components can be extracted from generally-unbounded infinitesimal generators. In conclusion the concept of module over a Banach algebra is proposed as the generalization of Banach algebra. As an application to mathematical physics, the rigorous formulation of rotation group, which consists of unbounded operators being written by differential operators, is provided using the module over a Banach algebra.
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The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution operators, where unboundedness of evolution operator is an essential ingredient of nonlinear analysis. In conclusion a general framework for the identification between the infinitesimal generators with evolution operators is established. A mathematical framework for such an identification is indispensable to the rigorous treatment of nonlinear transforms: e.g., transforms appearing in the theory of integrable systems.
Generally-unbounded infinitesimal generators are studied in the context of operator topology. Beginning with the definition of seminorm, the concept of locally convex topological vector space is introduced as well as the concept of Fr{e}chet space. These are the basic concepts for defining an operator topology. Consequently, by associating the topological concepts with the convergence of sequence, a suitable mathematical framework for obtaining the logarithmic representation of infinitesimal generators is presented.
Let $R := R_{2}(p)=mathbb{C}[t^{pm 1}, u : u^2 = t(t-alpha_1)cdots (t-alpha_{2n})] $ be the coordinate ring of a nonsingular hyperelliptic curve and let $mathfrak{g}otimes R$ be the corresponding current Lie algebra. color{black} Here $mathfrak g$ is a finite dimensional simple Lie algebra defined over $mathbb C$ and begin{equation*} p(t)= t(t-alpha_1)cdots (t-alpha_{2n})=sum_{k=1}^{2n+1}a_kt^k. end{equation*} In earlier work, Cox and Im gave a generator and relations description of the universal central extension of $mathfrak{g}otimes R$ in terms of certain families of polynomials $P_{k,i}$ and $Q_{k,i}$ and they described how the center $Omega_R/dR$ of this universal central extension decomposes into a direct sum of irreducible representations when the automorphism group was the cyclic group $C_{2k}$ or the dihedral group $D_{2k}$. We give examples of $2n$-tuples $(alpha_1,dots,alpha_{2n})$, which are the automorphism groups $mathbb G_n=text{Dic}_{n}$, $mathbb U_ncong D_n$ ($n$ odd), or $mathbb U_n$ ($n$ even) of the hyperelliptic curves begin{equation} S=mathbb{C}[t, u: u^2 = t(t-alpha_1)cdots (t-alpha_{2n})] end{equation} given in [CGLZ17]. In the work below, we describe this decomposition when the automorphism group is $mathbb U_n=D_n$, where $n$ is odd.
In this paper we {em discuss} diverse aspects of mutual relationship between adjoints and formal adjoints of unbounded operators bearing a matrix structure. We emphasize on the behaviour of row and column operators as they turn out to be the germs of an arbitrary matrix operator, providing most of the information about the latter {as it is the troublemaker}.
It is well-known that if T is a D_m-D_n bimodule map on the m by n complex matrices, then T is a Schur multiplier and $|T|_{cb}=|T|$. If n=2 and T is merely assumed to be a right D_2-module map, then we show that $|T|_{cb}=|T|$. However, this property fails if m>1 and n>2. For m>1 and n=3,4 or $ngeq m^2$, we give examples of maps T attaining the supremum C(m,n)=sup |T|_{cb} taken over the contractive, right D_n-module maps on M_{m,n}, we show that C(m,m^2)=sqrt{m} and succeed in finding sharp results for C(m,n) in certain other cases. As a consequence, if H is an infinite-dimensional Hilbert space and D is a masa in B(H), then there is a bounded right D-module map on the compact operators K(H) which is not completely bounded.
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