On the inverse of the generator of a bounded <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup

Let A be the generator of a uniformly bounded C ₀-semigroup in a Banach space B, and let A have a densely defined inverse A ^-1 . We present sufficient conditions on the resolvent (A-I ^-1, Re > 0, under which A ^-1 is also the generator of a uniformly bounded C ₀-semigroup.Dedicated to V. B. Lidskii on the occasion of his eightieth birthdayTranslated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 6–12, 2004Original Russian Text Copyright © by A. M. Gomilko     

On a criterion for the uniform boundedness of a <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup of operators in a Hilbert space

Let T(t), t ≥ 0, be a C ₀-semigroup of linear operators acting in a Hilbert space H with norm ‖·‖. We prove that T(t) is uniformly bounded, i.e., ‖T(t)‖ ≤ M, t ≥ 0, if and only if the following condition is satisfied:

Cayley transform of the generator of a uniformly bounded <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>-semigroup of operators

We consider the problem of estimates for the powers of the Cayley transform V = (A + I)(A - I)^–1 of the generator of a uniformly bounded C ₀-semigroup of operators e ^tA , t 0, that acts in a Hilbert space H. In particular, we establish the estimate . We show that the estimate is true in the following cases: (a) the semigroups e ^tA and are uniformly bounded; (b) the semigroup e ^tA uniformly bounded for t is analytic (in particular, if the generator of the semigroup is a bounded operator).Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1018–1029, August, 2004.     

Numerical Ranges of <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">N</Emphasis>) Contractions

A conjecture of Halmos proved by Choi and Li states that the closure of the numerical range of a contraction on a Hilbert space is the intersection of the closure of the numerical ranges of all its unitary dilations. We show that for C ₀(N) contractions one can restrict the intersection to a smaller family of dilations. This generalizes a finite dimensional result of Gau and Wu.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

The -conjecture and equivariant <Emphasis Type="Italic">e</Emphasis><Subscript><Emphasis Type="Italic">C</Emphasis></Subscript>-invariants

Let X be a smooth closed oriented non-spin 4-manifold with even intersection form kE₈nH (n1). The -conjecture states that n is greater than or equal to |k|. In this paper we give a proof of the -conjecture. The strategy of this paper is to use the finite dimensional approximation of the map induced from the Seiberg-Witten equations and equivariant e_C-invariants as in the paper of M. Furuta and Y. Kametani.Mathematics Subject Classification (1991): 57R55This work was supported by Korea Research Foundation Grant (KRF–2002–003–C00011).     

A generalized Kiepert formula for <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">ab</Emphasis></Subscript> curves

Kiepert (1873) and Brioschi (1864) published algebraic equations for the n-division points of an elliptic curve, in terms of the Weierstrass ℘-function and its derivatives with respect to a uniformizing parameter, or another elliptic function, respectively. We generalize both types of formulas for a compact Riemann surface which, outside from one point, has a smooth polynomial equation in the plane, in the sense that we characterize the points whose n-th multiple in the Jacobian belongs to the theta divisor.     

On the Theory of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>)-Banach Lattices

Given 1≤ p,q < ∞, let BL_pL_q be the class of all Banach lattices X such that X is isometrically lattice isomorphic to a band in some L_p(L_q)-Banach lattice. We show that the range of a positive contractive projection on any BL_pL_q-Banach lattice is itself in BL_pL_q. It is a consequence of this theorem and previous results that BL_pL_q is first-order axiomatizable in the language of Banach lattices. By studying the pavings of arbitrary BL_pL_q-Banach lattices by finite dimensional sublattices that are themselves in this class, we give an explicit set of axioms for BL_pL_q. We also consider the class of all sublattices of L_p(L_q)-Banach lattices; for this class (when p/q is not an integer) we give a set of axioms that are similar to Krivine’s well-known axioms for the subspaces of L_p-Banach spaces (when p/2 is not an integer). We also extend this result to the limiting case q = ∞.     

Similarity Results for Operators of Class <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>

If T is a multiplicity-free contraction of class C ₀ with minimal function m _T , then it is quasisimilar to the Jordan block S(m _T ). In case m _T is a Blaschke product with simple roots forming a Carleson sequence, we show that the relation between T and S(m _T ) can be strengthened to similarity. Under the additional assumption that u(T) has closed range for every inner divisor \({u\in H^\infty}\) of m _T , the result also holds in the more general setting where the roots have bounded multiplicities.     

Novikov superalgebras with <Emphasis Type="Italic">A</Emphasis><Subscript>0</Subscript> = <Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript><Emphasis Type="Italic">A</Emphasis><Subscript>1</Subscript>

Novikov superalgebras are related to quadratic conformal superalgebras which correspond to the Hamiltonian pairs and play a fundamental role in completely integrable systems. In this note we show that the Novikov superalgebras with A ₀ = A ₁ A ₁ and dim A ₁ = 2 are of type N and give a class of Novikov superalgebras of type S with A ₀ = A ₁ A ₁.     

A Class of <Emphasis Type="Italic">C</Emphasis>*-algebras with non-stable <Emphasis Type="Italic">K</Emphasis><Subscript>1</Subscript>-group property

In this paper, we give a class of C*-algebras with non-stable K ₁-group property which include the example non-simple tracial topological rank zero and stable rank two C*-algebra given by Lin and Osaka.     

Uniquely <Emphasis Type="Italic">C</Emphasis><Subscript>4</Subscript>-Saturated Graphs

For a fixed graph H, a graph G is uniquely H-saturated if G does not contain H, but the addition of any edge from [`(G)]{\overline{G}} to G completes exactly one copy of H. Using a combination of algebraic methods and counting arguments, we determine all the uniquely C ₄-saturated graphs; there are only ten of them.     

Recognition by spectrum of simple groups <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(2)

It is proved that, if G is a finite group that has the same set of element orders as the simple group C _p (2) for prime p > 3, then G/O ₂(G) is isomorphic to C _p (2).     

New results on <Emphasis Type="Italic">C</Emphasis><Subscript>11</Subscript> and <Emphasis Type="Italic">C</Emphasis><Subscript>12</Subscript> lattices with applications to Grothendieck categories and torsion theories

In this paper, which is a continuation of our previous paper [T. Albu, M. Iosif, A. Tercan, The conditions (C _i ) in modular lattices, and applications, J. Algebra Appl. 15 (2016), http: dx.doi.org/10.1142/S0219498816500018], we investigate the latticial counterparts of some results about modules satisfying the conditions (C ₁₁) or (C ₁₂). Applications are given to Grothendieck categories and module categories equipped with hereditary torsion theories.     

<Emphasis Type="Italic">l</Emphasis>-Groups <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) in continuous logic

In the context of continuous logic, this paper axiomatizes both the class \(\mathcal {C}\) of lattice-ordered groups isomorphic to C(X) for X compact and the subclass \(\mathcal {C}^+\) of structures existentially closed in \(\mathcal {C}\); shows that the theory of \(\mathcal {C}^+\) is \(\aleph _0\)-categorical and admits elimination of quantifiers; establishes a Nullstellensatz for \(\mathcal {C}\) and \(\mathcal {C}^+\); shows that \(C(X)\in \mathcal {C}\) has a prime-model extension in \(\mathcal {C}^+\) just in case X is Boolean; and proves that in a sense relevant to continuous logic, positive formulas admit in \(\mathcal {C}^+\) elimination of quantifiers to positive formulas.     

Continuous <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">p</Emphasis></Subscript>-symmetric distributions

For p > 0, the l _n,p -generalized surface measure on the l _n,p -unit sphere is studied and used for deriving a geometric measure representation for l _n,p -symmetric distributions having a density.     

<Emphasis Type="Italic">C</Emphasis>*-simplicity of <Emphasis Type="Italic">n</Emphasis>-periodic products

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.     

Smooth noncompact operators from <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">K</Emphasis>), <Emphasis Type="Italic">K</Emphasis> scattered

Let X be a Banach space, K be a scattered compact and T: B _C(K) → X be a Fréchet smooth operator whose derivative is uniformly continuous. We introduce the smooth biconjugate T**: B _C(K)** → X** and prove that if T is noncompact, then the derivative of T** at some point is a noncompact linear operator. Using this we conclude, among other things, that either is compact or that ℓ₁ is a complemented subspace of X*. We also give some relevant examples of smooth functions and operators, in particular, a C ^1,u -smooth noncompact operator from B _{c O} which does not fix any (affine) basic sequence. P. Hájek was supported by grants A100190502, Institutional Research Plan AV0Z10190503.     

<Emphasis Type="Italic">C</Emphasis>*-Homomorphisms and duality of <Emphasis Type="Italic">C</Emphasis>*-discrete quantum groups

Let

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be a C*-discrete quantum group and let

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be the discrete quantum group associated with

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. Suppose that there exists a continuous action of

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on a unital C*-algebra

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so that

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becomes a

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-algebra. If there is a faithful irreducible vacuum representation π of

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on a Hilbert space H =

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with a vacuum vector Ω, which gives rise to a

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-invariant state, then there is a unique C*-representation (θ, H) of

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supplemented by the action. The fixed point subspace of

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under the action of

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is exactly the commutant of θ(

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).