All operators considered in this paper are bounded and linear (everywhere defined) on a Hilbert space. An operator A will be called a square root of an operator B if

A simple sufficient condition guaranteeing that any solution A of (1) be normal whenever B is normal was obtained in [1], namely: If B is normal and if there exists some real angle θ for which Re(Aeιθ)≥0, then (1) implies that A is normal. Here, Re (C) denotes the real part ½(C + C*) of an operator C.

In a recent paper [1], the author divided the semi-special permutations on [n] that are not linear into two classes. The first class consists of the semi-special permutations which, for all possible values of s, have s as a principal number and which induce modulo s the identity permutation. The second class consists of all the semi-special permutations, with principal number s, which induce modulo s linear permutations other than the identity, where again s takes all its possible values.

In the course of some recent work on Fourier series [5, 6] I had occasion to use a number of integral inequalities which were generalizations or limiting cases of known results. These inequalities may perhaps have other applications, and it seems worth while to collect them together in a separate note with one or two further results of a similar nature.

For any number k, used as an index (exponent), and such that K > 1, we write k' = k′(k–1), so that k and k′ are conjugate indices in the sense of Hölder's inequality.

The following work establishes a new proof of the theorem: Every archimedean ordered group is abelian. This theorem has been proved differently by many authors. It was first proved by O. Hölder [2]. A second proof has been given by H. Cartan [1]: he uses the topology which is naturally introduced in the group by its order.

A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we write

The unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if

¶ y- z ¶≥ 2a

The statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.

In § 2 a number of infinite series of E-functions are summed by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration.

The Barnes integral employed is

where | amp z | < π and the integral is taken up the η;-axis, with loops, if necessary, to ensure that the origin lies to the left of the contour and the points α1, α2,… αp to the right of the contour. Zero and negative integral values of the α's and p's are excluded, and the α's must not differ by integral values. When p < q + 1 the contour is bent to the left at each end.

Given a series we define , by the relations

The series is said to be summable (C, k), where k > - 1, to the sum s if

to be summable (C, - 1) to s if it converges to s and nan = o(l); to be absolutely summable (C, k), or summable | C, k, to s if it is summable (C, k) to s and

and to be strongly Cesàro summable to s with order k > 0 and index p or summable [C; k, p] to s, if

Unless the contrary is stated, all matrices are understood to be complex and of type n × n. The transposed conjugate of A is denoted by A*. The non-negative square roots of the characteristic roots of A*A are called the singular values of A; they will be denoted by st(A), i = 1, …, n, where s1(A)≥…≥ sn(A). The symbol [A]k denotes the k × k submatrix standing in the upper left-hand corner of A. We shall write Ei(z1, …, zn) for the j-th elementary symmetric function of z1..., zn, and E1(A) for the j-th elementary symmetric function of the characteristic roots of A. It is understood that, throughout, 1≥j≥k≥n.

Many authors have proved results deducing an asymptotic expansion of

for large from the behaviour of f(t), when f(t) is regular in an appropriate part of the complex t-plane. For example, if, for some k > 0 and some Am, αm

for all large such that R(t) > C, then, as ⃗ ∞ in a suitable sector in the z-plane, we have

where Z is an appropriate value of z1/z.

Let Q be a complete local ring which has the same characteristic as its residue field P, and, for the present, let us denote by A the image of a subset A of Q under the natural homomorphism of Q onto P. Then a subfield F of Q is called a coefficient field if = P. It has been shown in [2] and in [3] that a complete equicharacteristic local ring, such as the above, always possesses at least one coefficient field; this is the embedding theorem for the equicharacteristic case.