e angles of the triangle are less than 120 ffi ; by the methods of elementary geometry. Let ABC b... more e angles of the triangle are less than 120 ffi ; by the methods of elementary geometry. Let ABC be a triangle all of whose angles are less than 120 ffi . Then the loci of interior points R such that "BRC = 120 ffi ; "CRA = 120 ffi ; "ARB = 120 ffi (1) are circular arcs that intersect at the required point F: This construction shows that there can be only one such point. It is equally clear that no such point can exist in a triangle when any angle is 120 ffi or more. Notation Let a; b; c be the position vectors of the vertices of the triangle ABC situated in the x; y 1 plane. For each point R(x; y) with position vector r define
. Suppose that h; k 2 L(H) are two selfadjoint bounded operators on a Hilbert space H. It is elem... more . Suppose that h; k 2 L(H) are two selfadjoint bounded operators on a Hilbert space H. It is elementary to show that hk is selfadjoint precisely when hk = kh. We answer the following question: under what circumstances must hk be selfadjoint given that it is normal? Throughout A will be a complex unital Banach algebra with unit e. Recall that an element a 2 A is hermitian if kexp(ita)k = 1 (t 2 R) and hermitian-equivalent if sup t2R kexp(ita)k ! 1: and that n is normal (-equivalent) if n = r + is where rs = sr and r; s are hermitian (-equivalent): we write n = r Gamma is for such an n. We shall use an extended two-normal Fuglede theorem: Theorem. Let A be a complex unital Banach algebra. If m; n (2 A) are normalequivalent, and na = am for some a 2 A, then n a = am . This follows from the fact that for any complex number we have a = e i n a e Gammai m so that 7! e in a e Gammaim = e i[ n+n ] a e Gammai[ m+m ] is bounded and analytic, hence c...
The Operator Kantorovich Inequality (R 2 Gamma r 2 ) u (a a) u R 2 (u a u)(u au) holds for a wide... more The Operator Kantorovich Inequality (R 2 Gamma r 2 ) u (a a) u R 2 (u a u)(u au) holds for a wide class of operators a on a Hilbert space H and all operators u : K ! H for which [a] u is a partial isometry, [a] being the range projection of a: 1991 Mathematics Subject Classification. Primary 47A63; Secondary 15A45, 65F65. Key words and phrases. Kantorovich Inequality, Cauchy-Schwarz Inequality. Introductory The Kantorovich Inequality 4mMOmega a Gamma1 ; ff (m +M) 2 kk 4 Omega a ; ff ; valid for a positive definite matrix with least eigenvalue m and largest eigenvalue M; (and any nonzero vector ) first appeared in [K] and has been generalized and applied extensively since. This Inequality may be viewed as a `reversal' of the special case Omega a ; ff ka k kk of the Cauchy-Schwarz Inequality, for it is equivalent to the inequality 2 p mM ka k kk (m +M)Omega a ; ff : The methods of Operator and Spectral Theory allow one to genera...
The Game of Diffy as described in [1], and there credited to H. Wills [2], is to choose a 4-word ... more The Game of Diffy as described in [1], and there credited to H. Wills [2], is to choose a 4-word (a set of four positive integers) and repeatedly form the (cyclic) absolute differences. (The game is won if this can be done six times before all the entries collapse to zero: our readers will always be able to win.) It has been shown that: 1 all 4-words eventually diff to zero; and 2 the number of steps before collapse can be arbitrarily large. The game can obviously be played with N-words, for any integer N ≥ 2. This raises the question: do all N-words diff to zero? The answer is yes when N = 2 n for some integer n; otherwise it is no. If N is not a power of 2 a word will eventually enter a cycle (which may by chance be trivial). There is no upper bound to the number of steps required to bring
The range of a contractive algebra morphism from a C∗-algebra to a Banach algebra is closed, and ... more The range of a contractive algebra morphism from a C∗-algebra to a Banach algebra is closed, and the morphism is a C∗-morphism onto its range. When the codomain is a dual Banach algebra, or only a right dual Banach algebra, such a morphism extends to a W∗-morphism onto the weak-star closure of the range (at least in the unital case). Boolean algebras of contractive projections (in right dual Banach algebras) have weak star completions; and operators with a contractive functional calculus on a dual Banach space are scalar type prespectral. Some of these results extend to morphisms that are neither unital nor contractive, so long as one can renorm the codomain dually in a suitable manner, as when the codomain is a dual Banach algebra, or when the range of the natural extension of the morphism forms a commuting set.
this paper is the interplay between order and lattice structure, commutativity, existence of squa... more this paper is the interplay between order and lattice structure, commutativity, existence of square roots, and monotonicity of powers, in the real part (the hermitian elements) of a unital hermitian Banach *algebra. These questions first appeared in the context of operators on Hilbert space. Lowner [Lo] established the monotonicity of the p
17P this way was 12320± 170 (S.E.M.). Each measurement requires only 0.4ml of solution; equilibri... more 17P this way was 12320± 170 (S.E.M.). Each measurement requires only 0.4ml of solution; equilibrium is reached within 20min.
The result stated in the title is a fundamental result about Banach algebras which appears to hav... more The result stated in the title is a fundamental result about Banach algebras which appears to have escaped notice up to now.
Proceedings of the Edinburgh Mathematical Society, 1975
Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a er-compl... more Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a er-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a tr-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of ^-completeness to weak relative compactness; indeed, a Boolean algebra of projections has a <x-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see ), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).
Proceedings of the Edinburgh Mathematical Society, 1972
The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded... more The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.
Proceedings of the American Mathematical Society, 1972
If H is the Hilbert transform on LP(Z), then T=ttI+ĂH is a well-bounded operator for \<p<ao, but ... more If H is the Hilbert transform on LP(Z), then T=ttI+ĂH is a well-bounded operator for \<p<ao, but is not a scalar-type spectral operator except when/>=2. The purpose of this note is to show that there is a well-bounded operator on a reflexive Banach space which is not scalar-type spectral.
We show that, given a weak compactness condition which is always satisfied when the underlying sp... more We show that, given a weak compactness condition which is always satisfied when the underlying space does not contain an isomorphic copy of c 0 , all the operators in the weakly closed algebra generated by the real and imaginary parts of a family of commuting scalar-type spectral operators on a Banach space will again be scalar-type spectral operators, provided that (and this is a necessary condition with even only two operators) the Boolean algebra of projections generated by their resolutions of the identity is uniformly bounded.
We present an algorithm which locates the poles and zeros of a rational function given the values... more We present an algorithm which locates the poles and zeros of a rational function given the values at the roots of unity, so long as enough values are speci ed to make the problem well-posed.
It is a classical theorem due to Kronecker that a Hankel operator with bounded measurable symbol ... more It is a classical theorem due to Kronecker that a Hankel operator with bounded measurable symbol on the unit circle has finite rank precisely when the antianalytic part of the symbol is rational. There is a similar result for the Hankel matrices analogously formed from the discrete Fourier transform of a continuous function: namely, that these matrices have uniformly bounded finite rank precisely when the symbol itself is rational.
It was shown by Bushel1 [l] that the equation t'rt= x2 has a unique positive-definite solution wh... more It was shown by Bushel1 [l] that the equation t'rt= x2 has a unique positive-definite solution when 1 is a real invertible matrix; the proof utilizes the Hilbert projective metric and the Banach fixed-point theorem. I present a simpler proof of a more general result.
e angles of the triangle are less than 120 ffi ; by the methods of elementary geometry. Let ABC b... more e angles of the triangle are less than 120 ffi ; by the methods of elementary geometry. Let ABC be a triangle all of whose angles are less than 120 ffi . Then the loci of interior points R such that "BRC = 120 ffi ; "CRA = 120 ffi ; "ARB = 120 ffi (1) are circular arcs that intersect at the required point F: This construction shows that there can be only one such point. It is equally clear that no such point can exist in a triangle when any angle is 120 ffi or more. Notation Let a; b; c be the position vectors of the vertices of the triangle ABC situated in the x; y 1 plane. For each point R(x; y) with position vector r define
. Suppose that h; k 2 L(H) are two selfadjoint bounded operators on a Hilbert space H. It is elem... more . Suppose that h; k 2 L(H) are two selfadjoint bounded operators on a Hilbert space H. It is elementary to show that hk is selfadjoint precisely when hk = kh. We answer the following question: under what circumstances must hk be selfadjoint given that it is normal? Throughout A will be a complex unital Banach algebra with unit e. Recall that an element a 2 A is hermitian if kexp(ita)k = 1 (t 2 R) and hermitian-equivalent if sup t2R kexp(ita)k ! 1: and that n is normal (-equivalent) if n = r + is where rs = sr and r; s are hermitian (-equivalent): we write n = r Gamma is for such an n. We shall use an extended two-normal Fuglede theorem: Theorem. Let A be a complex unital Banach algebra. If m; n (2 A) are normalequivalent, and na = am for some a 2 A, then n a = am . This follows from the fact that for any complex number we have a = e i n a e Gammai m so that 7! e in a e Gammaim = e i[ n+n ] a e Gammai[ m+m ] is bounded and analytic, hence c...
The Operator Kantorovich Inequality (R 2 Gamma r 2 ) u (a a) u R 2 (u a u)(u au) holds for a wide... more The Operator Kantorovich Inequality (R 2 Gamma r 2 ) u (a a) u R 2 (u a u)(u au) holds for a wide class of operators a on a Hilbert space H and all operators u : K ! H for which [a] u is a partial isometry, [a] being the range projection of a: 1991 Mathematics Subject Classification. Primary 47A63; Secondary 15A45, 65F65. Key words and phrases. Kantorovich Inequality, Cauchy-Schwarz Inequality. Introductory The Kantorovich Inequality 4mMOmega a Gamma1 ; ff (m +M) 2 kk 4 Omega a ; ff ; valid for a positive definite matrix with least eigenvalue m and largest eigenvalue M; (and any nonzero vector ) first appeared in [K] and has been generalized and applied extensively since. This Inequality may be viewed as a `reversal' of the special case Omega a ; ff ka k kk of the Cauchy-Schwarz Inequality, for it is equivalent to the inequality 2 p mM ka k kk (m +M)Omega a ; ff : The methods of Operator and Spectral Theory allow one to genera...
The Game of Diffy as described in [1], and there credited to H. Wills [2], is to choose a 4-word ... more The Game of Diffy as described in [1], and there credited to H. Wills [2], is to choose a 4-word (a set of four positive integers) and repeatedly form the (cyclic) absolute differences. (The game is won if this can be done six times before all the entries collapse to zero: our readers will always be able to win.) It has been shown that: 1 all 4-words eventually diff to zero; and 2 the number of steps before collapse can be arbitrarily large. The game can obviously be played with N-words, for any integer N ≥ 2. This raises the question: do all N-words diff to zero? The answer is yes when N = 2 n for some integer n; otherwise it is no. If N is not a power of 2 a word will eventually enter a cycle (which may by chance be trivial). There is no upper bound to the number of steps required to bring
The range of a contractive algebra morphism from a C∗-algebra to a Banach algebra is closed, and ... more The range of a contractive algebra morphism from a C∗-algebra to a Banach algebra is closed, and the morphism is a C∗-morphism onto its range. When the codomain is a dual Banach algebra, or only a right dual Banach algebra, such a morphism extends to a W∗-morphism onto the weak-star closure of the range (at least in the unital case). Boolean algebras of contractive projections (in right dual Banach algebras) have weak star completions; and operators with a contractive functional calculus on a dual Banach space are scalar type prespectral. Some of these results extend to morphisms that are neither unital nor contractive, so long as one can renorm the codomain dually in a suitable manner, as when the codomain is a dual Banach algebra, or when the range of the natural extension of the morphism forms a commuting set.
this paper is the interplay between order and lattice structure, commutativity, existence of squa... more this paper is the interplay between order and lattice structure, commutativity, existence of square roots, and monotonicity of powers, in the real part (the hermitian elements) of a unital hermitian Banach *algebra. These questions first appeared in the context of operators on Hilbert space. Lowner [Lo] established the monotonicity of the p
17P this way was 12320± 170 (S.E.M.). Each measurement requires only 0.4ml of solution; equilibri... more 17P this way was 12320± 170 (S.E.M.). Each measurement requires only 0.4ml of solution; equilibrium is reached within 20min.
The result stated in the title is a fundamental result about Banach algebras which appears to hav... more The result stated in the title is a fundamental result about Banach algebras which appears to have escaped notice up to now.
Proceedings of the Edinburgh Mathematical Society, 1975
Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a er-compl... more Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a er-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a tr-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of ^-completeness to weak relative compactness; indeed, a Boolean algebra of projections has a <x-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see ), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).
Proceedings of the Edinburgh Mathematical Society, 1972
The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded... more The notion of a well-bounded operator was introduced by Smart (9). The properties of well-bounded operators were further investigated by Ringrose (6, 7), Sills (8) and Berkson and Dowson (2). Berkson and Dowson have developed a more complete theory for the type (A) and type (B) well-bounded operators than is possible for the general well-bounded operator. Their work relies heavily on Sills' treatment of the Banach algebra structure of the second dual of the Banach algebra of absolutely continuous functions on a compact interval.
Proceedings of the American Mathematical Society, 1972
If H is the Hilbert transform on LP(Z), then T=ttI+ĂH is a well-bounded operator for \<p<ao, but ... more If H is the Hilbert transform on LP(Z), then T=ttI+ĂH is a well-bounded operator for \<p<ao, but is not a scalar-type spectral operator except when/>=2. The purpose of this note is to show that there is a well-bounded operator on a reflexive Banach space which is not scalar-type spectral.
We show that, given a weak compactness condition which is always satisfied when the underlying sp... more We show that, given a weak compactness condition which is always satisfied when the underlying space does not contain an isomorphic copy of c 0 , all the operators in the weakly closed algebra generated by the real and imaginary parts of a family of commuting scalar-type spectral operators on a Banach space will again be scalar-type spectral operators, provided that (and this is a necessary condition with even only two operators) the Boolean algebra of projections generated by their resolutions of the identity is uniformly bounded.
We present an algorithm which locates the poles and zeros of a rational function given the values... more We present an algorithm which locates the poles and zeros of a rational function given the values at the roots of unity, so long as enough values are speci ed to make the problem well-posed.
It is a classical theorem due to Kronecker that a Hankel operator with bounded measurable symbol ... more It is a classical theorem due to Kronecker that a Hankel operator with bounded measurable symbol on the unit circle has finite rank precisely when the antianalytic part of the symbol is rational. There is a similar result for the Hankel matrices analogously formed from the discrete Fourier transform of a continuous function: namely, that these matrices have uniformly bounded finite rank precisely when the symbol itself is rational.
It was shown by Bushel1 [l] that the equation t'rt= x2 has a unique positive-definite solution wh... more It was shown by Bushel1 [l] that the equation t'rt= x2 has a unique positive-definite solution when 1 is a real invertible matrix; the proof utilizes the Hilbert projective metric and the Banach fixed-point theorem. I present a simpler proof of a more general result.
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