共查询到20条相似文献,搜索用时 15 毫秒
1.
Let and denote respectively the space of n×n complex matrices and the real space of n×n hermitian matrices. Let p,q,n be positive integers such that p?q?n. For , the (p,q)-numerical range of A is the set , where Cp(X) is the pth compound matrix of X, and Jq is the matrix Iq?On-q. Let denote n or . The problem of determining all linear operators T: → such that is treated in this paper. 相似文献
2.
D.J Hartfiel 《Journal of Mathematical Analysis and Applications》1985,108(1):230-240
Let Pij and qij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where aij(t) is piecewise continuous, aij(t) = ?∑i ≠ jaij(t), and pij ? aij(t) ? qij all t. A solution x is also to satisfy ∑i = 1nxi(0) = 1. Let Ct denote the set of all solutions, evaluated at t to equations described above. It is shown that , the topological closure of Ct, is a compact convex set for each t. Further, the set valued function , of t is continuous and . 相似文献
3.
B.H Pourciau 《Journal of Mathematical Analysis and Applications》1982,87(2):373-381
Suppose Φ maps an open subset U of Rn into Rk, a?U, S is a subset of U, and int Φ(S) denotes the interior of the image Φ(S). Call any result with conclusion Φ(a) ? int Φ(S) an interior mapping theorem. The best known example is an easy corollary of the classical Implicit Mapping Theorem: if Φ is strongly differentiable at a?U and if L = Φ′(a), then Φ(a) ? int Φ(U) whenever L(a) ? int L(U), that is, whenever the linear transformation L maps Rn onto Rk. A more subtle interior mapping theorem is proved in this paper: if is a convex subset of U, if Φ is strongly differentiable at a?U, and if L = Φ′(a), then Φ(a) ? int Φ(C) whenever L(a) ? int L(C). This Convex Interior Mapping Theorem is then applied to yield a short proof of the Carathéodory-John Multiplier Rule for minimizing a strongly differentiable function φ0 subject to strongly differentiable inequality constraints φ1 ? 0,…, φp ? 0 strongly differentiable equality constraints φp + 1 = 0,…, φp + q= 0. (A corollary of this fundamental multiplier rule is the well-known Karush-Kuhn-Tucker theorem.) The demonstration proceeds by examining interiority properties of the mapping Φ = (φ0, φ1,…, φp + q) from U into Rp + q +1. 相似文献
4.
Gerhard Ramharter 《Journal of Number Theory》1982,14(2):269-279
For irrational numbers θ define α(θ) = lim sup{1/(q(p ? qθ))|p ∈ , q ∈ , p ? qθ > 0} and α(θ) = 0 for rationals. Put . Then = α(β) is an asymmetric analogue to the Lagrange spectrum . Our results concerning partly contrast the known properties of . In fact, is a perfect set, each element of which is a condensation point of the spectrum and has continuously many preimages. is the closure of its rational elements and of its elements of the form p√m (p ∈ ), as well. The arbitrarily well approximable numbers form a Gδ-set of 2. category. One has, roughly speaking, for α → 1. Finally, the well-known Markov sequence which constitutes the lower Lagrange and Markov spectrum is proved to be a (small) subset of ?[√5,3). 相似文献
5.
Milton Rosenberg 《Journal of multivariate analysis》1974,4(2):166-209
P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?” In this paper answers are given to the question, “When is a linear operator from q to p the integral of a spectral measure?”; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for q is the “inflation” of a spectral measure for . In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫(in0, 2π]e?iθE(dθ) is a unitary operator on and if T is a bounded linear operator from q to q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process (Unx)?∞∞ ?q (for each x = (xi)ij ∈ q, Unx = (Unxi)i=1q), then T is a spectral integral, ∫(0,2π)]Φ(θ) E(dθ), and the Banach norm of T, |T|B = ess sup |Φ(θ)|B. 相似文献
6.
A t-spread set [1] is a set of (t + 1) × (t + 1) matrices over GF(q) such that ∥C∥ = qt+1, 0 ? C, I?C, and det(X ? Y) ≠ 0 if X and Y are distinct elements of . The amount of computation involved in constructing t-spread sets is considerable, and the following construction technique reduces somewhat this computation. Construction: Let be a subgroup of GL(t + 1, q), (the non-singular (t + 1) × (t + 1) matrices over GF(q)), such that ∥G∥|at+1, and det (G ? H) ≠ 0 if G and H are distinct elements of . Let A1, A2, …, An?GL(t + 1, q) such that det(Ai ? G) ≠ 0 for i = 1, …, n and all G?G, and det(Ai ? AjG) ≠ 0 for i > j and all G?G. Let , and ∥C∥ = qt+1. Then is a t-spread set. A t-spread set can be used to define a left V ? W system over V(t + 1, q) as follows: x + y is the vector sum; let e?V(t + 1, q), then xoy = yM(x) where M(x) is the unique element of with x = eM(x). Theorem: Letbe a t-spread set and F the associatedV ? Wsystem; the left nucleus = {y | CM(y) = C}, and the middle nucleus = }y | M(y)C = C}. Theorem: Forconstructed as aboveG ? {M(x) | x?Nλ}. This construction technique has been applied to construct a V ? W system of order 25 with ∥Nλ∥ = 6, and ∥Nμ∥ = 4. This system coordinatizes a new projective plane. 相似文献
7.
Yoshimi Egawa 《Journal of Combinatorial Theory, Series A》1981,31(2):108-125
Distance-regular graphs which have the same parameters as the Hamming scheme H(n, q) are classified. If q ≠ 4, H(n, q) is the only such graph. If q = 4, there are precisely (isomorphism classes of) such graphs other than H(n, q). 相似文献
8.
Gordon S. Woodward 《Journal of Functional Analysis》1974,16(2):205-220
Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L1(G), C(G), and L∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C0(G) and Lp(G) for 1 < p < ∞ have such forms if and only if is a torsion group for some open σ-compact subgroup H of . For σ-compact amenable G's, all the above spaces have discontinuous left TILF's. 相似文献
9.
After the change of variables Δi = γi ? δi and xi,i + 1 = δi ? δi + 1 we show that the invariant polynomials μG(n)q(, Δi, ; , xi,i+1,) characterizing U(n) tensor operators 〈p, q,…, q, 0,…, 0〉 become an integral linear combination of Schur functions Sλ(γ ? δ) in the symbol γ ? δ, where γ ? δ denotes the difference of the two sets of variables {γ1 ,…, γn} and {δ1 ,…, δn}. We obtain a similar result for the yet more general bisymmetric polynomials mμG(n)q(γ1 ,…, γn; δ1 ,…, δm). Making use of properties of skew Schur functions and Sλ(γ ? δ) we put together an umbral calculus for mμG(n)q(γ; δ). That is, working entirely with polynomials, we uniquely determine mμG(n)q(γ; δ) from mμG(n)q ? 1(γ; δ) and combinatorial rules involving Ferrers diagrams (i.e., partitions), provided that n ≥ (μ + 1)q. (This restriction does not interfere with writing the general case of mμG(n)q(γ; δ) as a linear combination of Sλ(γ ? δ).) As an application we deduce “conjugation” symmetry for nμG(n)q(γ; δ) from “transposition” symmetry by showing that these two symmetries are equivalent. 相似文献
10.
Hermann König 《Journal of Functional Analysis》1977,24(1):32-51
For an open set Ω ? N, 1 ? p ? ∞ and λ ∈ +, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm (cf. A. Pietsch, “r-nukleare Sobol. Einbett. Oper., Ellipt. Dgln. II,” Akademie-Verlag, Berlin, 1971, pp. 203–215). Choose a Banach ideal of operators , 1 ? p, q ? ∞ and a quasibounded domain Ω ? N. Theorem 1 of the note gives sufficient conditions on λ such that the Sobolev-imbedding map exists and belongs to the given Banach ideal : Assume the quasibounded domain fulfills condition Ckl for some l > 0 and 1 ? k ? N. Roughly this means that the distance of any to the boundary ?Ω tends to zero as for , and that the boundary consists of sufficiently smooth ?(N ? k)-dimensional manifolds. Take, furthermore, 1 ? p, q ? ∞, p > k. Then, if μ, ν are real positive numbers with λ = μ + v ∈ , μ > λ S(; p,q:N) and v > N/l · λD(;p,q), one has that belongs to the Banach ideal . Here λD(;p,q;N)∈+ and λS(;p,q;N)∈+ are the D-limit order and S-limit order of the ideal , introduced by Pietsch in the above mentioned paper. These limit orders may be computed by estimating the ideal norms of the identity mappings lpn → lqn for n → ∞. Theorem 1 in this way generalizes results of R. A. Adams and C. Clark for the ideals of compact resp. Hilbert-Schmidt operators (p = q = 2) as well as results on imbeddings over bounded domains.Similar results over general unbounded domains are indicated for weighted Sobolev spaces.As an application, in Theorem 2 an estimate is given for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω fulfills condition C1l.For an open set Ω in N, let denote the Sobolev-Slobodetzkij space obtained by completing in the usual Sobolev-Slobodetzkij norm, see below. Taking a fixed Banach ideal of operators and 1 ? p, q ? ∞, we consider quasibounded domains Ω in N and give sufficient conditions on λ such that the Sobolev imbedding operator exists and belongs to the Banach ideal. This generalizes results of C. Clark and R. A. Adams for compact, respectively, Hilbert-Schmidt operators (p = q = 2) to general Banach ideals of operators, as well as results on imbeddings over bounded domains. Similar results over general unbounded domains may be proved for weighted Sobolev spaces. As an application, we give an estimate for the rate of growth of the eigenvalues of formally selfadjoint, uniformly strongly elliptic differential operators with Dirichlet boundary conditions in , where Ω is a quasibounded open set in N. 相似文献
11.
For 1 ? p ? ∞, let , be the lp norm of an m × n complex A = (αij) ?Cm × n. The main purpose of this paper is to find, for any p, q ? 1, the best (smallest) possible constants τ(m, k, n, p, q) and σ(m, k, n, p, q) for which inequalities of the form hold for all A?Cm × k, B?Ck × n. This leads to upper bounds for inner products on Ck and for ordinary lp operator norms on Cm × n. 相似文献
12.
Justin Peters 《Journal of Functional Analysis》1984,59(3):498-534
Given a C1-algebra and endomorphim α, there is an associated nonselfadjoint operator algebra + Xα, called the semi-crossed product of with α. If α is an automorphim, + Xα can be identified with a subalgebra of the C1-crossed product + Xα. If is commutative and α is an automorphim satisfying certain conditions, + Xα is an operator algebra of the type studied by Arveson and Josephson. Suppose S is a locally compact Hausdorff space, φ: S → S is a continuous and proper map, and α is the endomorphim of U=C0(S) given by α(?) = ? ō φ. Necessary and sufficient conditions on the map φ are given to insure that the semi-crossed product Z+XαC0(S) is (i) semiprime; (ii) semisimple; (ii) strongly semisimple. 相似文献
13.
A.L Carey 《Journal of Functional Analysis》1984,55(3):277-296
The group (H)2 of unitary operators (on a Hilbert space H) which differ from the identity by a Hilbert-Schmidt operator may be imbedded in the group of Bogoliubov automorphisms of the CAR algebra over H in such a way as to be weakly inner in any gauge-invariant quasifree representation. Consequently each such quasifree representation determines a projective representation of (H)2. If 0 ? A ? I is the operator on H determining the quasifree representation πA and ?A denotes the cyclic projective representation of (H)2 generated from the G.N.S. cyclic vector , then the 2-cocycle in (H)2 determined by ?A can be given explicitly. We prove that this 2-cocycle is a coboundary if any only if A or 1 ? A is Hilbert-Schmidt. The representations ?A, on restriction to the group (H)1 consisting of unitaries which differ from the identity by a trace class operator, always determine 2-cocycles which are coboundaries. These representations of (H)1 have already been investigated by 21., 22., 87–110). Thus the Stratila-Voiculescu representations of (H)1 always extend to projective representations of (H)2 and to ordinary representations when A or 1 ? A is Hilbert-Schmidt. This fact enables exploitation of the type analysis of Stratila and Voiculescu to determine the type of the von Neumann algebra ρA((H)2)″. In the special case where 0 and 1 are not eigenvalues of is cyclic and separating for ρA((H)2)″ and hence determines a K.M.S. state on this algebra. It is shown that for special choices of A, type IIIλ (0 < λ ? 1) factors ρA((H)2)″ may be constructed. 相似文献
14.
Stephen M. Paneitz 《Journal of Functional Analysis》1981,41(3):315-326
Let Sp() be the symplectic group for a complex Hibert space . Its Lie algebra sp() contains an open invariant convex cone C0; each element of C0 commutes with a unique sympletic complex structure. The Cayley transform : X∈ sp()→(I + X)1∈ Sp() is analyzed and compared with the exponential mapping. As an application we consider equations of the form is strongly continuous, and show that if ∝?∞∞ ∥A(t)∥ dt < 2 and ∝? t8∞A(t) dt?C0, the (scattering) operator , where St′(t) is the solution such that St′(t′) = I, is in the range of restricted to C0. It follows that S leaves invariant a unique complex structure; in particular, it is conjugate in Sp() to a unitary operator. 相似文献
15.
J.E Nymann 《Journal of Number Theory》1975,7(4):406-412
Given a set S of positive integers let denote the number of k-tuples 〈m1, …, mk〉 for which and (m1, …, mk) = 1. Also let denote the probability that k integers, chosen at random from , are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1 … pr) = 1}, then if k ≥ 3 and where d(S) denotes the natural density of S. From this result it follows immediately that as n → ∞. This result generalizes an earlier result of the author's where and S is then the whole set of positive integers. It is also shown that if S = {p1x1 … prxr : xi = 0, 1, 2,…}, then as n → ∞. 相似文献
16.
James F. Lynch 《Discrete Mathematics》1981,33(3):281-287
Two sets of sets, C0 and C1, are said to be visually equivalent if there is a 1-1 mapping m from C0 onto C1 such that for every S, T?C0, S ∩ T=0 if and only if m(S)∩ m(T)=0 and S?T if and only if m(S)?m(T). We find estimates for V(k), the number of equivalence classes of this relation on sets of k sets, for finite and infinite k. Our main results are that for finite k, , where α and β are approximately 0.7255 and 2.5323 respectively, and there is a set N of cardinality such that there are V(k) visually distinct sets of k subsets of N. 相似文献
17.
18.
Daniel J. Madden 《Journal of Number Theory》1978,10(3):303-323
If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions Kn over k(x) such that [K : k(x)] = pn using the concept of Witt vectors. This is accomplished in the following way; if [β1, β2,…, βn] is a Witt vector over k(x) = K0, then the Witt equation generates a tower of extensions through where . In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in Kn. This alternate generation has the form Ki = Ki?1(yi); yip ? yi = Bi, where, as a divisor in Ki?1, Bi has the form . In this form q is prime to Πpjλj and each λj is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field Kn over an algebraically closed field of constants. 相似文献
19.
The main concern of this paper is linear matrix equations with block-companion matrix coefficients. It is shown that general matrix equations AX ? XB = C and X ? AXB = C can be transformed to equations whose coefficients are block companion matrices: and , respectively, where ?L and CM stand for the first and second block-companion matrices of some monic r × r matrix polynomials L(λ) = λsI + Σs?1j=0λjLj and M(λ) = λtI + Σt7minus;1j=0λjMj. The solution of the equat with block companion coefficients is reduced to solving vector equations Sx = ?, where the matrix S is r2l × r2l[l = max(s, t)] and enjoys some symmetry properties. 相似文献
20.
Let us denote by R(k, ? λ)[R(k, ? λ)] the maximal number such that there exist different permutations of the set {1,…, k} such that any two of them have at least λ (at most λ, respectively) common positions. We prove the inequalities R(k, ? λ) ? kR(k ? 1, ? λ ? 1), R(k, ? λ) ? R(k, ? λ ? 1) ? k!, R(k, ? λ) ? kR(k ? 1, ? λ ? 1). We show: R(k, ? k ? 2) = 2, R(k, ? 1) = (k ? 1)!, R(pm, ? 2) = (pm ? 2)!, R(pm + 1, ? 3) = (pm ? 2)!, , R(k, ? 0) = k, R(pm, ? 1) = pm(pm ? 1), R(pm + 1, ? 2) = (pm + 1)pm(pm ? 1). The exact value of R(k, ? λ) is determined whenever k ? k0(k ? λ); we conjecture that R(k, ? λ) = (k ? λ)! for k ? k0(λ). Bounds for the general case are given and are used to determine that the minimum of |R(k, ? λ) ? R(k, ? λ)| is attained for . 相似文献