Minimal extension of a pseudoresolvent

In a Hilbert space we consider a pseudoresolvent whose range is the whole space. We introduce a set of associated operators which are compatible with the pseudoresolvent in the same way that an operator is compatible with its resolvent. For each associated operator we construct an extension whose resolvent is an extension of the pseudoresolvent and is minimal in a certain sense.Translated from Matematicheskie Zametki, Vol. 14, No. 1, pp. 95–99, July, 1973.The author wishes to thank V. É. Lyantse for his attention to this work.     

Integral closure of a quartic extension

Let R be a Noetherian unique factorization domain such that 2 and 3 are units,and let A=R[α]be a quartic extension over R by adding a rootαof an irreducible quartic polynomial p(z)=z4+az2+bz+c over R.We will compute explicitly the integral closure of A in its fraction field,which is based on a proper factorization of the coefficients and the algebraic invariants of p(z).In fact,we get the factorization by resolving the singularities of a plane curve defined by z4+a(x)z2+b(x)z+c(x)=0.The integral closure is expressed as a syzygy module and the syzygy equations are given explicitly.We compute also the ramifications of the integral closure over R.     

An extension of a Bourgain-Lindenstrauss-Milman inequality

Let ‖⋅‖ be a norm on Rn. Averaging ‖(ε₁x₁,…,εnxn)‖ over all the n² choices of , we obtain an expression |||x||| which is an unconditional norm on Rn. Bourgain, Lindenstrauss and Milman [J. Bourgain, J. Lindenstrauss, V.D. Milman, Minkowski sums and symmetrizations, in: Geometric Aspects of Functional Analysis (1986/1987), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 44-66] showed that, for a certain (large) constant η>1, one may average over ηn (random) choices of and obtain a norm that is isomorphic to |||⋅|||. We show that this is the case for any η>1.     

Order-continuous extension of a positive operator

For S. L. Sobolev's 80th birthday.     

An extension of a result of ribe

It is proved that for every 1≦p<∞, 1≦q<∞ and for every sequence {p _n}, 1≦p _n<∞,p _n→p, the spaceX=(Σ⊕l _{p n}) _q (resp.U=(Σ⊕L _{p n}(0, 1)) _q ) is uniformly homeomorphic toX⊕l _p (resp.U⊕L _p(0, 1)). This extends Ribe’s result from the casep=1 to generalp<∞ and thus provides examples of uniformly convex, uniformly homeomorphic Banach spaces which are not Lipschitz equivalent.     

An extension of a theorem of Kaplansky

A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky’s Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, for a division ring D of characteristic zero whose center intersects its multiplicative commutator group in a finite group, we prove that the counterpart of Kolchin’s Theorem over D implies that of Kaplansky’s Theorem over D. Next, we note that this proof, when adjusted in the setting of fields, provides a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that if Kaplansky’s Theorem holds over a division ring D, which is for instance the case over general fields, then a generalization of Kaplansky’s Theorem holds over D, and in particular over general fields.     

Uniqueness of extension of a regular measure

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 3, pp. 372–377, March, 1989.     

An extension of a theorem of Banach

Summary Via porosity a well-known theorem of Banach on bounded real valued continuous functions defined on [0, 1] is extended to higher dimensions. Summary. Via porosity a well-known theorem of Banach on bounded real valued continuous functions defined on [0, 1] is extended to higher dimensions.     

Largest extension of a finite convex geometry

We present a new embedding of a finite join-semidistributive lattice into a finite atomistic join-semidistributive lattice. This embedding turns out to be the largest extension, when applied to a finite convex geometry.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived September 18, 2002; accepted in final form September 29, 2003.     

On an extension of a combinatorial identity

Using Frobenius partitions we extend the main results of [4]. This leads to an infinite family of 4-way combinatorial identities. In some particular cases we get even 5-way combinatorial identities which give us four new combinatorial versions of Göllnitz-Gordon identities.     

Continuous extension of a densely parameterized semigroup

Let be a dense sub-semigroup of ℝ₊, and let X be a separable, reflexive Banach space. This note contains a proof that every weakly continuous contractive semigroup of operators on X over can be extended to a weakly continuous semigroup over ℝ₊. We obtain similar results for nonlinear, nonexpansive semigroups as well. As a corollary we characterize all densely parametrized semigroups which are extendable to semigroups over ℝ₊. O.M. Shalit was partially supported by the Gutwirth Fellowship.     

Extremal extension of a finitely additive measure

Translated from Matematicheskie Zametki, Vol. 43, No. 1, pp. 25–30, January, 1988.