Strongly quasibounded maximal monotone perturbations for the Berkovits-Mustonen topological degree theory

Let X be a real reflexive Banach space with dual X^∗. Let L:X⊃D(L)→X^∗ be densely defined, linear and maximal monotone. Let T:X⊃D(T)→X^∗2, with 0∈D(T) and 0∈T(0), be strongly quasibounded and maximal monotone, and C:X⊃D(C)→X^∗ bounded, demicontinuous and of type (S₊) w.r.t. D(L). A new topological degree theory has been developed for the sum L+T+C. This degree theory is an extension of the Berkovits-Mustonen theory (for T=0) and an improvement of the work of Addou and Mermri (for T:X→X^∗2 bounded). Unbounded maximal monotone operators with are strongly quasibounded and may be used with the new degree theory.     

Applications of the theory of semi-embeddings to Banach space theory

Let X and Y be Banach spaces and T:X → Y an injective bounded linear operator. T is called a semi-embedding if T maps the closed unit ball of X to a closed subset of Y. (This concept was introduced by Lotz, Peck, and Porta, Proc. Edinburgh Math. Soc.22 (1979), 233–240.) It is proved that if X semi-embeds in Y, and X is separable, then X has the Radon-Nikodym property provided Y does. It is shown that if L¹ semi-embeds in Y, then Y fails the Schur property and contains a subspace isomorphic to l¹. As a consequence of the proof, it is shown that if X is a subspace of L¹, either L¹ embeds in X or l¹ embeds in $\text{L}{}^1\text{X}$. The simpler result that L¹ does not semi-embed in c₀ is treated separately. This result is used to deduce the classic result of Menchoff that there exists a singular probability measure on the circle with Fourier coefficients vanishing at infinity. Some generalizations of the notion of semi-embedding are given, and several complements and open questions are discussed.     

A new topological degree theory for perturbations of the sum of two maximal monotone operators

Let X be an infinite dimensional real reflexive Banach space with dual space X^∗ and G⊂X, open and bounded. Assume that X and X^∗ are locally uniformly convex. Let T:X⊃D(T)→2^X∗ be maximal monotone and strongly quasibounded, S:X⊃D(S)→X^∗ maximal monotone, and C:X⊃D(C)→X^∗ strongly quasibounded w.r.t. S and such that it satisfies a generalized (S₊)-condition w.r.t. S. Assume that D(S)=L⊂D(T)∩D(C), where L is a dense subspace of X, and 0∈T(0),S(0)=0. A new topological degree theory is introduced for the sum T+S+C, with degree mapping d(T+S+C,G,0). The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S+C, as above.     

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X?D(T) → 2 ^X * is a maximal monotone multi-valued operator and C: X?D(C) → X* is a generalized pseudomonotone quasibounded operator with L ? D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T _x + λ C _x , with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.     

Multiplication operators in Köthe-Bochner spaces

Let X be a Banach space and E an order continuous Banach function space over a finite measure μ. We prove that an operator T in the Köthe-Bochner space E(X) is a multiplication operator (by a function in L^∞(μ)) if and only if the equality T(g〈f,x^∗〉x)=g〈T(f),x^∗〉x holds for every g∈L^∞(μ), f∈E(X), x∈X and x^∗∈X^∗.     

Products of longitudinal pseudodifferential operators on flag varieties

Associated to each set S of simple roots of SL(n,C) is an equivariant fibration X→XS of the complete flag variety X of Cn. To each such fibration we associate an algebra JS of operators on L²(X), or more generally on L²-sections of vector bundles over X. This ideal contains, in particular, the longitudinal pseudodifferential operators of negative order tangent to the fibres. Together, they form a lattice of operator ideals whose common intersection is the compact operators. Thus, for instance, the product of negative order pseudodifferential operators along the fibres of two such fibrations, X→XS and X→XT, is a compact operator if S∪T is the full set of simple roots. The construction of the ideals uses noncommutative harmonic analysis, and hinges upon a representation theoretic property of subgroups of SU(n), which may be described as ‘essential orthogonality of subrepresentations’.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.     

On the solution set of an equation of the type f(t, Φ(u(t)) = 0

We investigate the solution set Γ of an equation of the type f(t, Φ(u(t)) = 0, where Φ is a linear homeomorphism from a topological vector space X onto L ¹(T) and f: T×R → R is a Carathéodory function. More precisely, we characterize the property of Γ of intersecting each closed hyperplane of X.     

Drop properties and approximative compactness in Orlicz-Bochner function spaces

The representation of dual spaces of EM(μ,X), owing to its extensive application, is given in this paper. Using the representation, we get the sufficient and necessary conditions of LM(μ,X) possessing drop property, and extend the result of Hudzik and Wang [H. Hudzik, B. Wang, Approximative compactness in Orlicz spaces, J. Approx. Theory 95 (1998) 82-89]. Simultaneously, under some conditions, the weak^∗ drop property in LM(μ,X^∗) and LM^∗(μ,X) is discussed.     

Jordan-Ketten und realisierungen rationaler matrizen

If the inverse of a square polynomial matrix L(s) is proper rational, then L(s)^-1 can be written as C(sI–J)^-1B. The result of this note states that if J is an nXn Jordan matrix, with n=degreedetL(s), then C consists of Jordan chains of L(s), and B^T of Jordan chains of L(s)^T. This is a generalization of the fact that each matrix which transforms a complex matrix A into Jordan form is made up of eigenvectors and generalized eigenvectors of A. The proof of our result relies on the realization theory of rational matrices.     

Weyl's theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T∈B(X) is said to be totally hereditarily normaloid, T∈THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q?1, T∈H(q), if the quasi-nilpotent part H₀(T−λ)=(T−λ)^−q(0) for every complex number λ. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f(T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of σ(T), and T^∗ satisfies a-Weyl's theorem. If also T^∗ has the single valued extension property, then f(T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of σ(T) on which it is defined.     

Estimation of the Gauss–Markov process from observation of its sign

Let X(t) be the ergodic Gauss–Markov process with mean zero and covariance function e^?|τ|. Let D(t) be +1, 0 or ?1 according as X(t) is positive, zero or negative. We determine the non-linear estimator of X(t₁) based solely on D(t), ?T ? t ? 0, that has minimal mean–squared error ε²(t₁, T). We present formulae for ε²(t₁, T) and compare it numerically for a range of values of t₁ and T with the best linear estimator of X(t₁) based on the same data.     

The best generalized inverse of the linear operator in normed linear space

Let X,Y be normed linear spaces, T∈L(X,Y) be a bounded linear operator from X to Y. One wants to solve the linear problem Ax=y for x (given y∈Y), as well as one can. When A is invertible, the unique solution is x=A^-1y. If this is not the case, one seeks an approximate solution of the form x=By, where B is an operator from Y to X. Such B is called a generalised inverse of A. Unfortunately, in general normed linear spaces, such an approximate solution depends nonlinearly on y. We introduce the concept of bounded quasi-linear generalised inverse T^h of T, which contains the single-valued metric generalised inverse T^M and the continuous linear projector generalised inverse T⁺. If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalised inverses of T, denoted by G_H(T), is not empty In the normed linear space of all bounded homogeneous operators, the best bounded quasi-linear generalised inverse T^h of T is just the Moore-Penrose metric generalised inverse T^M. In the case, X and Y are finite dimension spaces Rⁿ and R^m, respectively, the results deduce the main result by G.R. Goldstein and J.A. Goldstein in 2000.     

Some properties of the injective tensor product of L[0,1] and a Banach space

Let X be a (real or complex) Banach space and 1<p,p′<∞ such that 1/p+1/p′=1. Then , the injective tensor product of L^p[0,1] and X, has the Radon-Nikodym property (resp. the analytic Radon-Nikodym property, the near Radon-Nikodym property, contains no copy of c₀, is weakly sequentially complete) if and only if X has the same property and each continuous linear operator from L^p′[0,1] to X is compact.     

Rank-1 perturbations of cosine functions and semigroups

Let A be the generator of a cosine function on a Banach space X. In many cases, for example if X is a UMD-space, A+B generates a cosine function for each B∈L(D((ω−A)^1/2),X). If A is unbounded and , then we show that there exists a rank-1 operator B∈L(D(γ(ω−A)),X) such that A+B does not generate a cosine function. The proof depends on a modification of a Baire argument due to Desch and Schappacher. It also allows us to prove the following. If A+B generates a distribution semigroup for each operator B∈L(D(A),X) of rank-1, then A generates a holomorphic C₀-semigroup. If A+B generates a C₀-semigroup for each operator B∈L(D(γ(ω−A)),X) of rank-1 where 0<γ<1, then the semigroup T generated by A is differentiable and ‖T^′(t)‖=O(t−α) as t↓0 for any α>1/γ. This is an approximate converse of a perturbation theorem for this class of semigroups.     

On normal control processes

Control processes of the form \(\dot x - A(t) x = B(t) u(t)\) , which are normal with respect to the unit ballB _{p′, r′} of the control spaceL ^p′([τ, T]),l _m ^r ′ are characterized in terms ofH(t)=X(T)X ^?1(t),B(t),X(t) any fundamental matrix solution of \(\dot x - A(t)x = 0\) , and directly in terms ofA, B, when bothA andB are independent oft.     

Discriminant loci of ample and spanned line bundles

Let (X,L,V) be a triplet where X is an irreducible smooth complex projective variety, L is an ample and spanned line bundle on X and V⊆H⁰(X,L) spans L. The discriminant locus D(X,V)⊂|V| is the algebraic subset of singular elements of |V|. We study the components of D(X,V) in connection with the jumping sets of (X,V), generalizing the classical biduality theorem. We also deal with the degree of the discriminant (codegree of (X,L,V)) giving some bounds on it and classifying curves and surfaces of codegree 2 and 3. We exclude the possibility for the codegree to be 1. Significant examples are provided.     

Nonlinear equations involving m-accretive operators

Let X be a Banach space and T an m-accretive operator defined on a subset D(T) of X and taking values in 2^x. For the class of spaces whose bounded closed and convex subsets have the fixed point property for nonexpansive self-mappings, it is shown here that two boundary conditions which imply existence of zeroes for T, appear to be equivalent. This fact is then used to prove that if there exists x₀?D(T) and a bounded open neighborhood U of x₀, such that $\text{¦T(x}{}_0\text{)¦ < r ? ¦T(x)¦}$ for all x??U ∩ D(T), then the open ball B(0; r) is contained in the range of T.     

Scalar conservation laws with general boundary condition and continuous flux function

We introduce a notion of entropy solution for a scalar conservation law on a bounded domain with nonhomogeneous boundary condition: ut+divΦ(u)=f on Q=(0,T)×Ω, u(0,⋅)=u₀ on Ω and “u=a on some part of the boundary (0,T)×∂Ω.” Existence and uniqueness of the entropy solution is established for any Φ∈C(R;RN), u₀∈L^∞(Ω), f∈L^∞(Q), a∈L^∞((0,T)×∂Ω). In the L¹-setting, a corresponding result is proved for the more general notion of renormalised entropy solution.