Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

Inverse Spectral Problems for Sturm-Liouville Equations with Eigenparameter Dependent Boundary Conditions

Inverse Sturm–Liouville problems with eigenparameter-dependentboundary conditions are considered. Theorems analogous to thoseof both Hochstadt and Gelfand and Levitan are proved. In particular, let ly = (1/r)(–(py')'+qy), , where det = > 0, c 0, det > 0, t 0 and (cs + dr –au – tb)² < 4(cr – ta)(ds – ub). Denoteby (l; ; ) the eigenvalue problem ly = y with boundary conditionsy(0)cos+y'(0)sin = 0 and (a+b)y(1) = (c+d)(py')(1). Define (; ; ) as above but with l replacedby . Let w_n denote the eigenfunctionof (l; ; ) having eigenvalue n and initial conditions w_n(0)= sin and pw'_n(0) = –cos and let _n = –aw_n(1)+cpw'_n(1).Define _n and _n similarly. As sample results, it is proved that if (l; ; ) and (; ; ) have the same spectrum, and (l;; ) and (; ; ) have the samespectrum or for all n, thenq/r = /.     

Interpolation of Vector-Valued Real Analytic Functions

Let R^d be an open domain. The sequentially complete DF-spacesE are characterized such that for each (some) discrete sequence(z_n) , a sequence of natural numbers (k_n) and any family the infinite system of equations has an E-valued real analytic solution f.     

Immanant Inequalities, Induced Characters, and Rank Two Partitions

If = {₁, ₂, ..., _s}, where _{1 2} ... _s > 0, is a partitionof n then denotes the associated irreducible character of S_n,the symmetric group on {1, 2, ..., n}, and, if cCS_n, the groupalgebra generated by C and S_n, then d_c(·) denotes thegeneralized matrix function associated with c. If c₁, c₂ CS_nthen we write c₁ c₂ in case (A) (A) for each n x n positivesemi-definite Hermitian matrix A. If cCS_n and c(e) 0, wheree denotes the identity in S_n, then or denotes (c(e))^–1 c. The main result, an estimate for the norms of tensors of a certainanti-symmetry type, implies that if = {₁, ₂, ..., _s, 1^t} isa partition of n such that s > 1 and _s = 2, and ' denotes{₁, ₂, ..., _s-1, 1^t} then (, _{2}) where denotes characterinduction from S_n–2 x S₂ to S_n. This in turn implies thatif = {₁, ₂, ..., _s, 1^t} with s > 1, _s = 2, and ßdenotes {₁ + 2, ₂, ..., _s-1, 1^t} then _ß which,in conjunction with other known results, provides many new inequalitiesamong immanants. In particular it implies that the permanentfunction dominates all normalized immanants whose associatedpartitions are of rank 2, a result which has proved elusivefor some years. We also consider the non-relationship problem for immanants– that is the problem of identifying pairs, (,ß)such that _ß and _ß are both false.     

On Artin's Braid Group And Polyconvexity In The Calculus Of Variations

Let R² be a bounded Lipschitz domain and let be a Carathèodory integrand such that F(x,·) is polyconvex for L²-a.e. x . Moreover assume thatF is bounded from below and satisfies the condition as det for L²-a.e. x . The paper describes the effect of domain topologyon the existence and multiplicity of strong local minimizersof the functional wherethe map u lies in the Sobolev space W_id^1,p (, R²) with p 2and satisfies the pointwise condition u(x) >0 for L²-a.e.x . The question is settled by establishing that F[·]admits a set of strong local minimizers on that can be indexed by the group P_n Zⁿ, the directsum of Artin's pure braid group on n strings and n copies ofthe infinite cyclic group. The dependence on the domain topologyis through the number of holes n in and the different mechanismsthat give rise to such local minimizers are fully exploitedby this particular representation.     

Periodic Solutions of Lienard Equations with Asymmetric Nonlinearities at Resonance

The existence of 2-periodic solutions of the second-order differentialequation where a, b satisfy and p(t)=p(t+2),t R, is examined. Assume that limits lim_x±F(x)=F(±)(F(x)=) and lim_x±g(x)=g(±)exist and are finite. It is proved that the equation has atleast one 2-periodic solution provided that the zeros of thefunction ₁ are simple and the zeros of the functions ₁, ₂ aredifferent and the signs of ₂ at the zeros of ₁ in [0,2/n) donot change or change more than two times, where ₁ and ₂ aredefined as follows: Moreover, it is also proved that the given equation has at leastone 2-periodic solution provided that the following conditionshold: with 1 p < q 2.     

Weighted Integral Inequalities for the Hardy Type Operator and the FractionalMaximal Operator

Let w(x), u(x) and (x) be weight functions. In this paper, underappropriate conditions on Young's functions ₁, ₂ we characterizethe inequality for the Hardy-typeoperator T defined in [1] and the inequality for the fractional maximal operator M_{, ;} definedin [8], as well as the corresponding weak-type inequalities.     

On the L1 Mean of the Exponential Sum Formed with the Mobius Function

In this paper we study the L¹ mean (1) of the exponential sum M()=_nXµ(n)e(n), where µ(n)is the Möbius function and e(x)=e^2ix. From the Cauchy–Schwarzinequality and Parseval's identity, we have , (2) and it is an interesting problem to investigate whether (2)reflects the true order of magnitude of (1).     

Tangential Boundary Behaviour of Harmonic and Holomorphic Functions

Let K be a kernel on Rⁿ, that is, K is a non-negative, unboundedL¹ function that is radially symmetric and decreasing. We definethe convolution K * F by and note from L^p-capacity theory [11, Theorem 3] that, if F L^p, p > 1, then K * F exists as a finite Lebesgue integraloutside a set A Rⁿ with C_K,p(A) = 0. For a Borel set A, where We define the Poisson kernel for = {(x, y) : x Rⁿ, y > 0} by and set Thus u is the Poisson integral of the potential f = K * F, andwe write u=P_y*(K*F)=P_y*f=P[f]. We are concerned here with the limiting behaviour of such harmonicfunctions at boundary points of , and in particular with the tangential boundary behaviour ofthese functions, outside exceptional sets of capacity zero orHausdorff content zero.     

Turan's Inequalities for Trigonometric Polynomials

We present here a technique for establishing inequalities ofthe form in the set of alltrigonometric polynomials of order n which have only real zeros.The function is assumed to be convex and increasing on [0,). As a corollary of the main result we get Turan's inequalities with the exact constantc(n, k, q) for each 1 q , n and k.     

Normal Elements of C*-Algebras of Real Rank Zero without Finite-Spectrum Approximants

We investigate inductive limits of Toeplitz-type C*-algebras.One example, which has real-rank zero, is the middle term ofan exact sequence where is a Bunce-Deddens algebra and I is AF. Using Berg's technique,we produce a normal element N that is not the limit of finite-spectrumnormals. Moreover, this is an example of a normal element inan inductive limit that is not the limit of normal elementsof the approximating subalgebras. A second example is an embedding of C() ( the closed disk) into , where is a simple AF algebra and is the Toeplitz algebra.Let _n, for n 2, be the CW complex obtained as the quotientof by an n-fold identification of the boundary. (So ₂ = RP².)Regarding C(_n) as a subalgebra of C(), we find nontrivial embeddingsof C(_n) into type I inductive limits. From this, we producea *-homomorphism, for n odd, C₀(_n\{pt}) _{n + 1}, that inducesan isomorphism on K-theory. More generally, for X a connectedCW complex minus a point, and for n odd, we show that the map is a split surjection.     

An Entire Function Defined by a Nonlinear Recurrence Relation

For the nonlinear recurrence relation it is proved that the limit exists and defines an entire function of ₂ = ₁(1-₁).     

Ergodicity of a Class of Cocycles Over Irrational Rotations

It is proved that if is irrational and L²(S¹) with o(l/n)then for each mZ\{0} the corresponding skew product is ergodic. The rigidity of specialflows over irrational rotations with roof functions whose Fouriercoefficients are in o(l/n) is also shown.     

On the Length of Snakes in Powers of Complete Graphs

The conjecture stated in an earlier paper by the author thatthere is a constant (independent from both n and k) such that n^d–1 holds for everyn 2 and d 2, where is the length of the longest snake (cycle without chords) in the Cartesianproduct of d copies of thecomplete graph K_n, is proved.     

Merit Factors of Character Polynomials

Let q be a prime and be a non-principal character modulo q.Let where 1 t q is the character polynomial associated to (cyclicallypermuted t places). The principal result is that for any non-principaland non-real character modulo q and 1 t q, where the implicit constant is independent of t and q. Here||·||₄ denotes the L₄ norm on the unit circle. It follows from this that all cyclically permuted characterpolynomials associated with non-principal and non-real charactershave merit factors that approach 3. This complements and completesresults of Golay, Høholdt and Jensen, and Turyn (andothers). These results show that the merit factors of cyclicallypermuted character polynomials associated with non-principalreal characters vary asymptotically between 3/2 and 6. The averages of the L₄ norms are also computed. Let q be a primenumber. Then where the summation is over all characters modulo q.     

On the Index of Vector Fields Tangent to Hypersurfaces with Non-Isolated Singularities

Let F be a germ of a holomorphic function at 0 in Cⁿ⁺¹, having0 as a critical point not necessarily isolated, and let be a germ of a holomorphic vectorfield at 0 in Cⁿ⁺¹ with an isolated zero at 0, and tangent toV := F^–1(0). Consider the O_V,0-complex obtained by contractingthe germs of Kähler differential forms of V at 0 (0.1) with the vector field X:=|_Von V: (0.2)     

Asymptotic Expansions of Multiple Zeta Functions and Power Mean Values of Hurwitz Zeta Functions

Let (s, ) be the Hurwitz zeta function with parameter . Powermean values of the form are studied, where q and h are positive integers. These mean valuescan be written as linear combinations of , where _r(s₁,...,s_r;) is a generalization of Euler–Zagiermultiple zeta sums. The Mellin–Barnes integral formulais used to prove an asymptotic expansion of , with respect to q. Hence a general way of deducingasymptotic expansion formulas for is obtained. In particular, the asymptotic expansion of with respect to q is written down.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

UNEXPECTED SUBSPACES OF TENSOR PRODUCTS

We describe complemented copies of ₂ both in C(K₁) C(K₂) when at least one of the compact spaces K_iis not scattered and in L₁(µ₁)L₁(µ₂) when at least one of the measures is not atomic.The corresponding local construction gives uniformly complementedcopies of the in c₀ c₀. We continue the study of c₀ c₀ showing that it contains a complementedcopy of Stegall's space and proving that (c₀ c₀)' is isomorphicto , together with other results. In the last section we use Hardy spaces to find an isomorphiccopy of L_p in the space of compact operators from L_q to L_r,where 1 < p, q, r < and 1/r = 1/p + 1/q.