Subspaces Of Finite Codimension Research Articles

Structure of strongly proximinal subspaces

In this paper we extended some results of G. Godefroy, originally proved for c0 using its special geometry, to the class of complex Banach spaces X whose dual is isometric to L1(μ). We exhibit classes of complex Banach spaces in which every strongly proximinal finite codimensional subspace is a finite dimensional perturbation of a M-ideal. We also show for this class, when the unit ball has an extreme point, if every proximinal subspace of codimension one is strongly proximinal, then the space is isometric to the k-dimensional space with the maximum norm, ℓ∞(k).

A Hilbert space approach to singularities of functions

We introduce the notion of a pseudomultiplier of a Hilbert space H of functions on a set Ω. Roughly, a pseudomultiplier of H is a function which multiplies a finite-codimensional subspace of H into H, where we allow the possibility that a pseudomultiplier is not defined on all of Ω. A pseudomultiplier of H has singularities, which comprise a subspace of H, and generalize the concept of singularities of an analytic function, even though the elements of H need not enjoy any sort of analyticity. We analyse the natures of these singularities, and obtain a broad classification of them in function-theoretic terms.

Minimality of eigenfunctions and associated functions of ordinary differential operators

Let E, F, H be Banach spaces, $$A,B\in L(E,F)$$ such that $$A+\lambda B$$ and its adjoint $$A^*+\lambda B^*$$ have a biorthogonal system of eigenvectors and associated vectors $$(y_i)_{i=1}^\infty$$ and $$(v_i)_{i=1}^\infty$$ . It is assumed that a closed finite codimensional subspace $$E_0$$ of E exists such that $$E_0$$ is dense in H and such that the restriction of B to $$E_0$$ has a continuous extension to H. It is shown that a subsystem $$(y_i)_{i=k+1}^\infty$$ is minimal in H if $$(v_i)_{i=1}^k$$ has a certain basis property. This abstract result can be applied to regular ordinary linear differential operators in Sobolev spaces whose boundary conditions may depend linearly on the eigenvalue parameter for proving minimality in $$L_p$$ . Explicit examples for Sturm–Liouville problems with one or both boundary conditions depending on $$\lambda$$ are considered.

On proximinality of subspaces and the lineability of the set of norm-attaining functionals of Banach spaces

We show that for every 1<n<∞, there exists a Banach space Xn containing proximinal subspaces of codimension n but no proximinal finite codimensional subspaces of higher codimension. Moreover, the set of norm-attaining functionals of Xn contains n-dimensional subspaces, but no subspace of higher dimension. This gives an n-by-n version of the solutions given by Read and Rmoutil to problems of Singer and Godefroy. We also study the existence of strongly proximinal subspaces of finite codimension, showing that for every 1<n<∞ and 1⩽k<n, there is a Banach space Xn,k containing proximinal subspaces of finite codimension up to n but not higher, and containing strongly proximinal subspaces of finite codimension up to k but not higher. Finally, we deal with possible infinite-dimensional versions of the previous results showing, among other results, that there are non-separable Banach spaces whose set of norm-attaining functionals contains infinite-dimensional separable linear subspaces but no non-separable subspaces.

A Cheney and Wulbert type lifting theorem in optimization

In this paper we exhibit new classes of Banach spaces for which strong notions of optimization can be lifted from quotient spaces. Motivated by a well known result of Cheney and Wulbert on lifting of proximinality from a quotient space to a subspace, for closed subspaces, \(Z \subset Y \subset X\), we consider stronger forms of optimization, that Z has in X and the quotient space Y / Z has in X / Z should lead to the conclusion Y has the same property in X. The versions we consider have been studied under various names in the literature as L-proximinal subspaces or subspaces that have the strong-\(1\frac{1}{2}\)-ball property. We give an example where the strong-\(1\frac{1}{2}\)-ball property fails to lift to the quotient. We show that if every M-ideal in Y is a M-summand, for a finite codimensional subspace \(Z \subset Y\), that is a M-ideal in X with the strong-\(1\frac{1}{2}\)-ball property in X and if Y / Z has the \(1\frac{1}{2}\)-ball property in X / Z, then Y has the strong-\(1\frac{1}{2}\)-ball property in X.

Complex Exceptional Orthogonal Polynomials and Quasi-invariance

Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1, \dots, \lambda_n)$ is a partition. G\'omez-Ullate et al showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of $\mathbb C[x]$ satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schr\"odinger operators, is also presented.

On the Bishop–Phelps–Bollobás theorem for operators and numerical radius

We study the Bishop–Phelps–Bollobas property for numerical radius (for short, BPBp-nu) of operators on `1-sums and `∞-sums of Banach spaces. More precisely, we introduce a property of Banach spaces, which we call strongly lush. We find that if X is strongly lush and X ⊕1 Y has the weak BPBp-nu, then (X,Y ) has the Bishop–Phelps– Bollobas property (BPBp). On the other hand, if Y is strongly lush and X ⊕∞ Y has the weak BPBp-nu, then (X,Y ) has the BPBp. Examples of strongly lush spaces are C(K) spaces, L1(μ) spaces, and finite-codimensional subspaces of C[0, 1].

Equivalence after extension for compact operators on Banach spaces

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In the present paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions for compact Hilbert space operators to be equivalent after extension, in terms of their singular values, we prove, under certain additional conditions, the necessity of a similar relationship between the s-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary s-functions. We investigate equivalence after extension for operators on ℓp-spaces. We show that two operators that act on different ℓp-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension; however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces X and Y, we investigate consequences of an operator on X being equivalent after extension to a compact operator on Y. We show that, in this case, a closed finite codimensional subspace of Y must embed into X, and that certain general Banach space properties must transfer from X to Y. We also show that no operator on X can be equivalent after extension to an operator on Y, if X and Y are essentially incomparable Banach spaces.

C k -solvability near the characteristic set for a class of elliptic vector fields with degeneracies

This paper deals with the solvability near the characteristic set \({\Sigma = \{0\} \times S^{1}}\) of operators of the form \({L= \partial / \partial t + (x^{n}a(x) + ixb(x))\partial / \partial x, b(0) \neq 0 \,\,{\rm and}\,\, n \geq 2, \,\,{\rm defined \,\, on}\,\, \Omega_{\epsilon} = (-\epsilon,\epsilon) \times S^{1}, \epsilon > 0,}\) where a and b are real-valued smooth functions in \({(-\epsilon,\epsilon). \,\,{\rm For \,\, fixed}\,\, k \geq 1}\), it is shown that given f belonging to a subspace of finite codimension (depending on k) of \({C^{\infty}(\Omega_{\epsilon})}\) there is a solution \({u \in C^{k}}\) of the equation Lu = f in a neighborhood of \({\Sigma}\).

Strong proximinality and intersection properties of balls in Banach spaces

We investigate a variation of the transitivity problem for proximinality properties of subspaces and intersection properties of balls in Banach spaces. For instance, we prove that if Z⊆Y⊆X, where Z is a finite co-dimensional subspace of X which is strongly proximinal in Y and Y is an M-ideal in X, then Z is strongly proximinal in X. Towards this, we prove that a finite co-dimensional proximinal subspace Y of X is strongly proximinal in X if and only if Y⊥⊥ is strongly proximinal in X⁎⁎. We also prove that in an abstract L1-space, the notions of strongly subdifferentiable points and quasi-polyhedral points coincide. We also give an example to show that M-ideals need not be ball proximinal. Moreover, we prove that in an L1-predual space, M-ideals are ball proximinal.

The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces. This property has an influence in the non-Archimedean Grothendieck’s approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces.

A Strongly Convergent Combined Relaxation Method in Hilbert Spaces

We consider a combined relaxation method for variational inequalities in a Hilbert space setting. Methods of this class are known to solve finite-dimensional variational inequalities under mild monotonicity type assumptions, whereas in Hilbert space strong monotonicity is the standard assumption for strong convergence. Here, we relax this condition and show strong convergence of such a method, when strong monotonicity holds only on a subspace of finite co-dimension. Thus, the method applies to semi-coercive unilateral boundary value problems in mathematical physics.

Finite codimensional invariant subspace and uniform algebra

Finite codimensional invariant subspaces in the abstract Hardy space defined by a unique representing measure on a uniform algebra are described.

On the Problem of Distance From a Point to the Subspace of Finite Codimension in Real Banach Space

In this paper,assuming X is Banach and L is a finite codimension subspace of X. Application of the dual method,there obtain the formula of distance and elements of best approximation from given a point to the subspace of finite codimension under some condition,and apply the formula to solve some of the specific optimal control problem.

Three-space problems for the bounded compact approximation property

National Natural Science Foundation of China [10526034, 10701063]; Fundamental Research Funds for the Central Universities [2011121039]; NSF [DMS-0800061, DMS-1068838]

Design of a Composite Membrane with Patches

This paper is concerned with minimization and maximization problems of eigenvalues. The principal eigenvalue of a differential operator is minimized or maximized over a set which is formed by intersecting a rearrangement class with an affine subspace of finite co-dimension. A solution represents an optimal design of a 2-dimensional composite membrane Ω, fixed at the boundary, built out of two different materials, where certain prescribed regions (patches) in Ω are occupied by both materials. We prove existence results, and present some features of optimal solutions. The special case of one patch is treated in detail.

Banach Lie algebras with Lie subalgebras of finite codimension have Lie ideals

This paper continues the work started in [E. Kissin, V. S. Shulman and Yu. V. Turovskii, �Banach Lie algebras with Lie subalgebras of finite codimension: their invariant subspaces and Lie ideals�, J. Funct. Anal. 256 (2009) 323�351.] and is devoted to the study of reducibility of an infinite-dimensional Lie algebra of operators on a Banach space when its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. In addition to the tools developed in the above paper; filtrations of Banach spaces with respect to Lie algebras of operators and related systems of operators on graded Banach spaces, the present paper introduces and studies some new concepts and techniques: the theory of Lie quasi-ideals and properties of Lie nilpotent finite-dimensional subspaces of Banach associative algebras. The application of these techniques to an operator Lie algebra L shows that, under some mild additional assumptions, L is reducible if its Lie subalgebra of finite codimension has an invariant subspace of finite codimension. This, in turn, leads to the main result of the paper: if a Banach Lie algebra L has a closed Lie subalgebra of finite codimension, then it has a proper closed Lie ideal of finite codimension. Moreover, if L is non-commutative, then it has a characteristic Lie ideal of finite codimension, that is, a proper closed Lie ideal of L invariant for all bounded derivations of L.

Densely ball remotal subspaces of C(K)

We call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K) **.

Projections of normed linear spaces with closed subspaces of finite codimension as kernels

It follows from [1], [4] and [7] that any closed ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> n$-codimensional subspace ($n \ge 1$ integer) of a real Banach space $X$ is the kernel of a projection $X \to X$, of norm less than $f(n) + \varepsilon$~($\varepsilon > 0$ arbitrary), where \[ f (n) = \frac{2 + (n-1) \sqrt{n+2}}{n+1}. \] We have $f(n) 1$, and \[ f(n) = \sqrt{n} - \frac{1}{\sqrt{n}} + O \left(\frac{1}{n}\right). \] (The same statement, with $\sqrt{n}$ rather than $f(n)$, has been proved in [2]. A~small improvement of the statement of [2], for $n = 2$, is given in [3], pp.~61--62, Remark.) In [1] for this theorem a deeper statement is used, on approximations of finite rank projections on the dual space $X^*$ by adjoints of finite rank projections on $X$. In this paper we show that the first cited result is an immediate consequence of the principle of local reflexivity, and of the result from [7].

A Remark on non-separable solutions to the Schroeder-Bernstein Problem for Banach spaces

We show that the geometric structure of Banach spaces which are solutions to the Schroeder-Bernstein Problem is very complex. More precisely, we prove that there exists a non-separable solution E to this problem such that (a) E is isomorphic to each one of its finite codimensional subspaces. (b) E has no complemented Hereditarily Indecomposable subspace. (c) E has no complemented subspace isomorphic to its square. (d) E has no non-trivial divisor.