Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E?F are symmetric sequence spaces with E≠?₁ and F≠c₀ and ?_∞ then there exist a intermediate symmetric sequence space G such that E?G?F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c₀ and ?₁ inside the class of all symmetric sequence spaces are given.     

Type and infratype in symmetric sequence spaces

We construct a symmetric sequence space that is of infratype 2 but not of type 2. Dedicated to J. Lindenstrauss. Work partially supported by an NSF grant.     

Twistor fibrations over Hermitian symmetric spaces and harmonic maps

Given a twistor space over a Hermitian symmetric space of compact type we construct a map onto a twistor space over another inner symmetric space of compact type. This map is holomorphic and preserves the superhorizontal distributions. We describe an application to harmonic maps.     

Distances between certain symmetric spaces

This paper is devoted to estimates of Banach-Mazur distances between finite-dimensional symmetric spaces. Lower and upper estimates are obtained in terms of fundamental sequences of symmetric spaces.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 113, pp. 218–224, 1981.     

Equilibrium maps between metric spaces

We show the existence of harmonic mappings with values in possibly singular and not necessarily locally compact complete metric length spaces of nonpositive curvature in the sense of Alexandrov. As a technical tool, we show that any bounded sequence in such a space has a subsequence whose mean values converge. We also give a general definition of harmonic maps between metric spaces based on mean value properties and-convergence.     

On symmetric basic sequences in Lorentz sequence spaces

We examine the symmetric basic sequences in some classes of Banach spaces with symmetric bases. We show that the Lorentz sequence spaced(a,p) has a unique symmetric basis and every infinite dimensional subspace ofd(a,p) contains a subspace isomorphic tol ^p. The symmetric basic sequences ind(a,p) are identified and a necessary and sufficient condition for a Lorents sequence space with exactly two nonequivalent symmetric basic sequences in given. We conclude by exhibiting an example of a Lorentz sequence space having a subspace with symmetric basis which is not isomorphic either to a Lorentz sequence space or to anl ^p-space. This is part of the first author's Ph. D. thesis, prepared at the Hebrew University of Jerusalem under the supervision of Dr. L. Tzafriri.     

q-concavity and related properties on symmetric sequence spaces

We introduce a new property between the q-concavity and the lower q-estimate of a Banach lattice and we get a general method to construct maximal symmetric sequence spaces that satisfies this new property but fails to be q-concave. In particular this gives examples of spaces with the Orlicz property but without cotype 2.     

Orlicz property and cotype in symmetric sequence spaces

We construct a symmetric sequence space that satisfies the Orlicz property but that fails to be of cotype 2. Work partially supported by an NSF grant.     

Nonlinear Markov operators associated with symmetric Markov kernels and energy minimizing maps between singular spaces

We present a theory of harmonic maps where the target is a complete geodesic space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov and the domain is a measure space with a symmetric Markov kernel p on it. Our theory is a nonlinear generalization of the theory of symmetric Markov kernels and reversible Markov chains on M. It can also be regarded as a particular case of the theory of generalized (= nonlinear) Dirichlet forms and energy minimizing maps between singular spaces, initiated by Jost (1994) and Korevaar, Schoen (1993) and developed further by Jost (1997a), (1998) and Sturm (1997). We investigate the discrete and continuous time heat flow generated by p and show that various properties of the linear heat flow carry over to this nonlinear heat flow. In particular, we study harmonic maps, i.e. maps which are invariant under the heat flow. These maps are identified with the minimizers of the energy. Received April 2, 2000 / Accepted May 9, 2000 /Published online November 9, 2000     

Sobolev spaces and harmonic maps between singular spaces

Recently Korevaar and Schoen developed a Sobolev theory for maps from smooth (at least ) manifolds into general metric spaces by proving that the weak limit of appropriate average difference quotients is well behaved. Here we extend this theory to functions defined over Lipschitz manifold. As an application we then prove an existence theorem for harmonic maps from Lipschitz manifolds to NPC metric spaces. Received December 6, 1996 / Accepted March 4, 1997     

Decomposable symmetric mappings between infinite-dimensional spaces

Decomposable mappings from the space of symmetric k-fold tensors over E, , to the space of k-fold tensors over F, , are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.     

The Banach-Mazur distance between symmetric spaces

We show that the Banach-Mazur distance betweenN-dimensional symmetric spacesE andF satisfies , wherec is a numerical constant. IfE is a symmetric space, then max (M ⁽²⁾(E),M ₍₂₎(E)), whereM ⁽²⁾(E) (resp.M ₍₂₎(E)) denotes the 2-convexity (resp. the 2-concavity) constant ofE. We also give an example of a spaceF with an 1-unconditional basis and enough symmetries that satisfiesd(F, l ₂ ^dimF )=M ⁽²⁾(F)M ₍₂₎(F). Partially supported by NSF Grant MCS-8201044.     

A spectral identity between symmetric spaces

Let π be a cuspidal automorphic representation ofGL _2n . We prove an identity between two spectral distributions onSp _2n andGL _2n respectively. The first is the spherical distribution with respect toSp _n×Sp _nof the residual Eisenstein series induced from π. The second is the weighted spherical distribution of π with respect toGL _n×GL _nand a certain degenerate Eisenstein series. A similar identity relates the pair (U _2n ,Sp _n) and (GL _n/E,GL _n/F) whereE/F is the quadratic extension defining the quasi-split unitary groupU _2n . We also have a Whittaker version of these trace identities. First-named author partially supported by NSF grant DMS 0070611. Second-named author partially supported by NSF grant DMS 9970342.     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.