scholarly journals Operators with Powers close to a Fixed Operator

1969 ◽  
Vol 9 (1-2) ◽  
pp. 237-238
Author(s):  
Mary R. Embry
Keyword(s):  
Positive Integer ◽  
Linear Operator ◽  
Real Number ◽  
Complex Number ◽  
Continuous Linear Operator ◽  
Continuous Linear ◽  
Image Position ◽  
Positive Integers

It is intuitively obvious that if z is a complex number such that ∣1–zv∣ ≤ b ≺ 1 for all positive integers p and some real number b, then z = 1. The purpose of this note is to exhibit a proof of the following generalisation of this observation: THEOREM. Let A be continuous linear operator on a reflexive Banachspace B. If there exists a continuous linear operator T on B, a real number b, and a positive integer p' such that, p an integer and , then A = I. Moreover, in this case ∥I—T∥.

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

On eigenvalues and inverse singular values of compact linear operators in Hilbert space

1953 ◽  
Vol 49 (2) ◽  
pp. 201-212 ◽  
Author(s):  
J. P. O. Silberstein ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Continuous Linear Operator ◽  
Singular Values ◽  
Linear Operators ◽  
Complex Numbers ◽  
Identity Operator ◽  
Continuous Linear ◽  
Completely Continuous ◽  
Image Position

1·1. In this paper we shall be concerned with the equationswhere K is a compact (completely continuous) linear operator in a Hilbert space , K is the adjoint of K, I is the identity operator, x and y are elements of ∥ x ∥ denotes the norm of x, and κ and σ are complex numbers.

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

A Combinatorial Theorem

1966 ◽  
Vol 9 (4) ◽  
pp. 515-516
Author(s):  
Paul G. Bassett
Keyword(s):  
Positive Integer ◽  
Combinatorial Theorem ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Positive Integers

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:1Define2In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

scholarly journals Semi-embeddings of Banach spaces which are hereditarily c0

1983 ◽  
Vol 26 (2) ◽  
pp. 163-167 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Banach Spaces ◽  
Continuous Linear Operator ◽  
Closed Unit Ball ◽  
Continuous Linear ◽  
Scattered Space ◽  
Complete Set

Following Lotz, Peck and Porta [9], a continuous linear operator from one Banach space into another is called a semi-embedding if it is one-to-one and maps the closed unit ball of the domain onto a closed (hence complete) set. (Below we shall allow the codomain to be an F-space, i.e., a complete metrisable topological vector space.) One of the main results established in [9] is that if X is a compact scattered space, then every semi-embedding of C(X) into another Banach space is an isomorphism ([9], Main Theorem, (a)⇒(b)).

EXOTIC LAPLACIANS AND DERIVATIVES OF WHITE NOISE

2011 ◽  
Vol 14 (01) ◽  
pp. 1-14 ◽  
Author(s):  
LUIGI ACCARDI ◽  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
Linear Operator ◽  
White Noise ◽  
Continuous Linear Operator ◽  
Higher Order ◽  
Continuous Linear ◽  
White Noise Functionals ◽  
Derivatives Of

In this paper, we give a relationship between the exotic Laplacians and the Lévy Laplacians in terms of the higher-order derivatives of white noise by introducing a bijective and continuous linear operator acting on white noise functionals. Moreover, we study a relationship between exotic Laplacians, acting on higher-order singular functionals, each other in terms of the constructed operator.

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

scholarly journals A note on Dunford-Pettis operators

1987 ◽  
Vol 29 (2) ◽  
pp. 271-273 ◽  
Author(s):  
J. R. Holub
Keyword(s):  
Linear Operator ◽  
Positive Operator ◽  
Continuous Linear Operator ◽  
Positive Operators ◽  
Regular Operator ◽  
Continuous Linear

Talagrand has shown [4, p. 76] that there exists a continuous linear operator from L1[0, 1] to c0 which is not a Dunford-Pettis operator. In contrast to this result, Gretsky and Ostroy [2] have recently proved that every positive operator from L[0, 1] to c0 is a Dunford-Pettis operator, hence that every regular operator between these spaces (i.e. a difference of positive operators) is Dunford-Pettis.

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