bluuebunny/crossref_metadata_2025 huggingface dataset:
Equations with constant coefficients invariant under a group of linear transformations
Authors: Cerezo André
Published: July 2010

If

P
P

is a linear differential operator on

R

n

{{\mathbf {R}}^n}

with constant coefficients, which is invariant under a group

G
G

of linear transformations, it is not true in general that the equation

P
u
=
f

Pu = f

always has a

G
G

-invariant solution

u
u

for a

G
G

-invariant

f
f

. We elucidate here the particular case of a “big” group

G
G

, and we count the invariant solutions when they exist (see Corollary 28 and Theorems 32, 33). The case, of special interest, of the wave equation and the Lorentz group is covered (Corollary 27). The theory of hyperfunctions provides the frame for the work.
Müntz-Szász theorem with integral coefficients. II
Authors: Ferguson Le Baron O. , von Golitschek Manfred
Published: July 2010

The classical Müntz-Szász theorem concerns uniform approximation on [0, 1] by polynomials whose exponents are taken from a sequence of real numbers. Under mild restrictions on the exponents or the interval, the theorem remains valid when the coefficients of the polynomials are taken from the integers.
Analytic structure in some analytic function algebras
Authors: Zame William R.
Published: July 2010

A complete description is given of the analytic structure of maximal dimension in the spectra of a wide class of concrete function algebras generated by analytic functions. A connection is also given with point derivations on such algebras.
The quasi-orbit space of continuous 𝐶*-dynamical systems
Authors: Takai Hiroshi
Published: July 2010

Let

(
A
,
G
,
α
)

(A,G,\alpha )

be a separable continuous

C
∗

{C^\ast }

-dynamical system. Suppose G is amenable and

α
\alpha

is free on the dual Â of A. Then the quasi-orbit space

(

Prim

A

/

α

)
∼

{({\text {Prim}}\;A/\alpha )^ \sim }

of the primitive ideal space Prim A of A by

α
\alpha

is homeomorphic to the induced primitive ideal space which is dense in the primitive ideal space Prim

C
∗

(
A
;
α
)

{C^\ast }(A;\alpha )

of the

C
∗

{C^\ast }

-crossed product

C
∗

(
A
;
α
)

{C^\ast }(A;\alpha )

of A by

α
\alpha

.
A product formula for generalizations of the Kervaire invariant
Authors: Brown Edgar H.
Published: July 2010

Formulas are developed for the Arf invariant of the product of two manifolds in terms of invariants of the factors. If the Wu orientations are carefully chosen the formula is

σ
(
M
×
N
)
=
σ
(
M
)
σ
(
N
)

\sigma (M \times N) = \sigma (M)\sigma (N)

.
On the topological extension to the boundary of biholomorphic maps in 𝐶ⁿ
Authors: Range R. Michael
Published: July 2010

Let

F
:

D
1

→

D
2

F:{D_1} \to {D_2}

be a biholomorphic map between bounded domains in

C

n

{{\mathbf {C}}^n}

with piecewise smooth strictly pseudoconvex boundaries. It is shown that F is Hölder continuous of some positive order, and hence F extends to a homeomorphism of the closures of the domains. This generalizes recent results of G. M. Henkin and N. Vormoor for domains with smooth strictly pseudoconvex boundary.
Geometry of Banach spaces of functions associated with concave functions
Authors: Hlavac Paul , Sundaresan K.
Published: July 2010

Let

(
X
,
Σ
,
μ
)

(X,\Sigma ,\mu )

be a positive measure space, and

ϕ
\phi

be a concave nondecreasing function on

R
+

→

R
+

{R^ + } \to {R^ + }

with

ϕ
(
0
)
=
0

\phi (0) = 0

. Let

N
ϕ

(
R
)

{N_\phi }(R)

be the Lorentz space associated with the function

ϕ
\phi

. In this paper a complete characterization of the extreme points of the unit ball of

N
ϕ

(
R
)

{N_\phi }(R)

is provided. It is also shown that the space

N
ϕ

(
R
)

{N_\phi }(R)

is not reflexive in all nontrivial cases, thus generalizing a result of Lorentz. Several analytical properties of spaces

N
ϕ

(
R
)

{N_\phi }(R)

, and their abstract analogues

N
ϕ

(
E
)

{N_\phi }(E)

, are obtained when E is a Banach space.
A generalization of H. Weyl’s “unitary trick”
Authors: Lepowsky J.
Published: July 2010

H. Weyl’s “unitary trick” is generalized to the context of semisimple symmetric Lie algebras with Cartan subspaces, over fields of characteristic zero. As an illustration of its usefulness, the result is used to transfer to characteristic zero an important theorem in the representation theory of real semisimple Lie algebras.
The semilattice tensor product of distributive lattices
Authors: Fraser Grant A.
Published: July 2010

We define the tensor product

A
⊗
B

A \otimes B

for arbitrary semilattices A and B. The construction is analogous to one used in ring theory (see [4], [7], [8]) and different from one studied by A. Waterman [12], D. Mowat [9], and Z. Shmuely [10]. We show that the semilattice

A
⊗
B

A \otimes B

is a distributive lattice whenever A and B are distributive lattices, and we investigate the relationship between the Stone space of

A
⊗
B

A \otimes B

and the Stone spaces of the factors A and B. We conclude with some results concerning tensor products that are projective in the category of distributive lattices.
Homogeneous manifolds with negative curvature. I
Authors: Azencott Robert , Wilson Edward N.
Published: July 2010

This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature. Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra

s

\mathfrak {s}

satisfying these conditions and any member of an easily constructed family of inner products on

s

\mathfrak {s}

, a metric deformation argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group.
Light open and open mappings on manifolds. II
Authors: Walsh John J.
Published: July 2010

Sufficient conditions are given for the existence of light open mappings between p.l. manifolds. In addition, it is shown that a mapping f from a p.l. manifold

M
m

,
m
⩾
3

{M^m},m \geqslant 3

, to a polyhedron Q is homotopic to an open mapping of M onto Q iff the index of

f
∗

(

π
1

(
M
)
)

{f_\ast }({\pi _1}(M))

in

π
1

(
Q
)

{\pi _1}(Q)

is finite. Finally, it is shown that an open mapping of

M
m

{M^m}

onto a p.l. manifold

N
n

,
n
⩾
m
⩾
3

{N^n},n \geqslant m \geqslant 3

, can be approximated by a light open mapping of M onto N.
The sets that are scissor congruent to an unbounded convex subset of the plane
Authors: Gold Sydell Perlmutter
Published: July 2010

It is shown that an unbounded convex plane body is scissor congruent to the union of a congruent body with a finite number of arbitrary topological discs. It is proved that ’is scissor congruent to’ is an equivalence relation. Thus two unbounded convex plane bodies are scissor congruent if and only if the union of one with a finite number of topological discs is scissor congruent to the other.
Certain continua in 𝑆ⁿ of the same shape have homeomorphic complements
Authors: Liem Vo Thanh
Published: July 2010

As a consequence of Theorem 1 of this paper, we see that if X and Y are globally 1-alg continua in

S
n

(
n
⩾
5
)

{S^n}\;(n \geqslant 5)

having the shape of the real projective space

P
k

(
k
≠
2
,
2
k
+
2
⩽
n
)

{P^k}\;(k \ne 2,2k + 2 \leqslant n)

, then

S
n

−
X
≈

S
n

−
Y

{S^n} - X \approx {S^n} - Y

. (For

P
1

=

S
1

{P^1} = {S^1}

, this establishes the last case of such a result for spheres.) We also show that if X and Y are globally 1-alg continua in

S
n

,
n
⩾
6

{S^n},n \geqslant 6

, which have the shape of a codimension

⩾
3

\geqslant 3

, closed,

0
>
(
2
m
−
n
+
1
)

0 > (2m - n + 1)

-connected, PL-manifold

M
m

{M^m}

, then

S
n

−
X
≈

S
n

−
Y

{S^n} - X \approx {S^n} - Y

.
Cylindric algebras of first-order languages
Authors: Myers Dale
Published: July 2010

We show when two countable first-order languages have isomorphic cylindric algebras.
Cohomology of finite covers
Authors: Calder Allan
Published: July 2010

For a finite dimensional CW-complex, X, and

q
>
1

q > 1

, it is shown that the qth Čech cohomology group based on finite open covers of X,

H
f
q

(
X
)

H_f^q(X)

, is naturally isomorphic to

H
q

(
X
)

{H^q}(X)

, the qth Čech cohomology of X (i.e. based on locally finite covers), and for reasonable X,

H
1

(
X
)

{H^1}(X)

can be obtained algebraically from

H
f
1

(
X
)

H_f^1(X)

.
Hermite series as boundary values
Authors: Walter G. G.
Published: July 2010

The relations between Hermite series expansions of functions and tempered distributions on the real axis and holomorphic or harmonic functions or generalizations of them in the upper half plane are studied. The Hermite series expansions of

H
2

{H^2}

functions are characterized in terms of their coefficients. Series of analytic representations of Hermite functions, series of Hermite functions of the second kind, and combined series of Hermite functions of the first and second kind are investigated. The functions to which these series converge in the upper half plane are shown to approach (in various ways) the distributions or functions whose Hermite series have the same coefficients.
Analytic hypoellipticity of certain second-order evolution equations with double characteristics
Authors: Tosques Mario
Published: July 2010

The present article establishes the analytic hypoellipticity (Definition 1.2) of a class of abstract evolution equations of order two, with double characteristics, under the hypothesis that the coefficients are analytic (in a suitable sense; see §2). The noteworthy feature of the main result (Theorem 4.1) is that analytic hypoellipticity holds whenever hypoellipticity does, even when one of the asymptotic eigenvalues

c
j

(
A
)

{c^j}(A)

fails to be elliptic of order one.
Hyperfinite extensions of bounded operators on a separable Hilbert space
Authors: Moore L. C.
Published: July 2010

Let H be a separable Hilbert space and Ĥ the nonstandard hull of H with respect to an

ℵ
1

{\aleph _1}

-saturated enlargement. Let S be a

∗

^\ast

-finite dimensional subspace of

∗

H

^\ast H

such that the corresponding hyperfinite dimensional subspace Ŝ of Ĥ contains H. If T is a bounded operator on H, then an extension Â of T to Ŝ where Â is obtained from an internal

∗

^\ast

-linear operator on S is called a hyperfinite extension of T. It is shown that T has a compact (selfadjoint) hyperfinite extension if and only if T is compact (selfadjoint). However T has a normal hyperfinite extension if and only if T is subnormal. The spectrum of a hyperfinite extension Â equals the point spectrum of Â, and if T is quasitriangular, A can be chosen so that the spectrum of Â equals the spectrum of T. A simple proof of the spectral theorem for bounded selfadjoint operators is given using a hyperfinite extension.
Concordances of Hilbert cube manifolds
Authors: Chapman T. A.
Published: July 2010

The main result of this paper asserts that homotopy groups of concordances of compact Hilbert cube manifolds are isomorphic to homotopy groups of concordances of compact finite-dimensional piecewise-linear manifolds. This enables us to apply some finite-dimensional results to obtain some new information about homotopy groups of homeomorphism groups of compact Hilbert cube manifolds. Our approach also yields a much shorter proof of the local contractibility of the homeomorphism group of any compact Hilbert cube manifold.
Characteristic classes for the deformation of flat connections
Authors: Lue Huei Shyong
Published: July 2010

In this paper, we study the secondary characteristic classes derived from flat connections. Let M be a differential manifold with flat connection

ω
0

{\omega _0}

. If f is a diffeomorphism of M, then

ω
1

=

f
∗

ω
0

{\omega _1} = {f^\ast }{\omega _0}

is another flat connection. Denote by

α
\alpha

the difference of these two connections. Then

α
\alpha

and its exterior covariant derivative

D
α

D\alpha

are both tensorial forms on M. To each invariant polynomial

φ
\varphi

of

GL

(
n
,

R

)

{\text {GL}}(n,{\text {R}})

, where

n
=
dim
⁡
M
,
φ
(
α
;
D
α
)

n = \dim M,\varphi (\alpha ;D\alpha )

is a globally defined form on M. The class

{
φ
(
α
;
D
α
)
}
∈
H
(
M
;

R

)

\{ \varphi (\alpha ;D\alpha )\} \in H(M;{\text {R}})

for

deg
⁡
φ
>
1

\deg \varphi > 1

gives rise to an obstruction of the deformability from

ω
0

{\omega _0}

to

ω
1

{\omega _1}

. In particular, we prove that

(
+
)

( + )

and

(
−
)

( - )

connections, in the sense of E. Cartan, cannot be deformed to each other.
Universally torsionless and trace modules
Authors: Garfinkel Gerald S.
Published: July 2010

We study, over an arbitrary ring R, a class of right modules intermediate between the projective and the flat content modules. Over the ring of rational integers these modules are the locally free abelian groups. Over any commutative ring they are the modules which remain torsionless under all scalar extensions. They each possess a certain separability property exactly when R is left semihereditary. We define M to be universally torsionless if the natural map

M
⊗
A
→

Hom

(

M
∗

,
A
)

M \otimes A \to {\operatorname {Hom}}({M^\ast },A)

is monic for all left modules A. We give various equivalent conditions for M to be universally torsionless, one of which is that M is a trace module, i.e. that

x
∈
M
⋅

M
∗

(
x
)

x \in M \cdot {M^\ast }(x)

for all

x
∈
M

x \in M

. We show the countably generated such modules are projective. Chase showed that rings over which products of projective or flat modules are also, respectively, projective or flat have other interesting properties and that they are characterized by certain left ideal theoretical conditions. We show similar results hold when the trace or content properties are preserved by products.
The Čech cohomology of movable and 𝑛-movable spaces
Authors: Keesling James
Published: July 2010

In this paper the Čech cohomology of movable and n-movable spaces is studied. Let X be a space and let

H
k

(
X
)

{H^k}(X)

denote the k-dimensional Čech cohomology of X with integer coefficients based on the numerable covers of X. Then if X is movable, there is a subgroup E of

H
k

(
X
)

{H^k}(X)

which is the union of all the algebraically compact subgroups of

H
k

(
X
)

{H^k}(X)

. Furthermore,

H
k

(
X
)

/

E

{H^k}(X)/E

is an

ℵ
1

{\aleph _1}

-free abelian group. If X is an n-movable space, then it is shown that this structure holds for

H
k

(
X
)

{H^k}(X)

for

0
⩽
k
⩽
n

0 \leqslant k \leqslant n

and may be false for

k
⩾
n
+
1

k \geqslant n + 1

. If X is an

LC

n
−
1

{\text {LC}^{n - 1}}

paracompactum, then X is known to be n-movable. However, in this case and in the case that X is an

LC

n
−
1

{\text {LC}^{n - 1}}

compactum a stronger structure theorem is proved for

H
k

(
X
)

{H^k}(X)

for

0
⩽
k
⩽
n
−
1

0 \leqslant k \leqslant n - 1

than that stated above. Using these results examples are given of n-movable continua that are not shape equivalent to any

LC

n
−
1

{\text {LC}^{n - 1}}

paracompactum.
The multiplicative behavior of ℋ
Authors: Grillet Pierre Antoine
Published: July 2010

Various results are given describing the product of two

H

\mathcal {H}

-classes in an arbitrary semigroup in terms of groups and homomorphisms.
On the connectedness of homomorphisms in topological dynamics
Authors: McMahon D. , Wu T. S.
Published: July 2010

Let (X, T) be a minimal transformation group with compact Hausdorff phase space. We show that if

ϕ
:
(
X
,
T
)
→
(
Y
,
T
)

\phi :(X,T) \to (Y,T)

is a distal homomorphism and has a structure similar to the structure Furstenberg derived for distal minimal sets, then for T belonging to a class of topological groups T, the homomorphism

X
→
X

/

S
(
ϕ
)

X \to X/S(\phi )

has connected fibers, where

S
(
ϕ
)

S(\phi )

is the relativized equicontinuous structure relation. The class T is defined by Sacker and Sell as consisting of all groups T with the property that there is a compact set

K
⊆
T

K \subseteq T

such that T is generated by each open neighborhood of K. They show that for such T, a distal minimal set which is a finite-to-one extension of an almost periodic minimal set is itself an almost periodic minimal set. We provide an example that shows that the restriction on T cannot be dropped. As one of the preliminaries to the above we show that given

ϕ
:
(
X
,
T
)
→
(
Y
,
T
)

\phi :(X,T) \to (Y,T)

, the relation

R
c
(
ϕ
)

Rc(\phi )

induced by the components in the fibers relative to

ϕ
\phi

, i.e.,

(
x
,

x
′

)
∈
R
c
(
ϕ
)

(x,x’) \in Rc(\phi )

if and only if x and

x
′

x’

are in the same component of

ϕ

−
1

(
ϕ
(
x
)
)

{\phi ^{ - 1}}(\phi (x))

, is a closed invariant equivalence relation. We also consider the question of when a minimal set (X, T) is such that

Q
(
x
)

Q(x)

is finite for some x in X, where Q is the regionally proximal relation. This problem was motivated by Veech’s work on almost automorphic minimal sets, i.e., the case in which

Q
(
x
)

Q(x)

is a singleton for some x in X.
Extreme points of univalent functions with two fixed points
Authors: Silverman Herb
Published: July 2010

Univalent functions of the form

f
(
z
)
=

a
1

z
−

Σ

n
=
2

∞

a
n

z
n

f(z) = {a_1}z - \Sigma _{n = 2}^\infty {a_n}{z^n}

, where

a
n

⩾
0

{a_n} \geqslant 0

, are considered. We examine the subclasses for which

f
(

z
0

)
=

z
0

f({z_0}) = {z_0}

or

f
′

(

z
0

)
=
1
,

z
0

f’({z_0}) = 1,{z_0}

real. The extreme points of these classes that are starlike of order

α
\alpha

are determined.
On the zeros of Dirichlet 𝐿-functions. III
Authors: Fujii Akio
Published: July 2010

It is shown that the ordinates of the zeros of the Riemann zeta function are uniformly distributed. Similar results pertain to zeros of L-functions.
Differential games with Lipschitz control functions and applications to games with partial differential equations
Authors: Barron Emmanuel Nicholas
Published: July 2010

In §1 we formulate a differential game when the dynamics is the inhomogeneous heat equation. In §2 we state the basic theory of differential games when the controls must choose uniformly Lipschitz control functions. We then prove some general theorems for the case when the controls may choose any measurable control functions. These theorems hold for games with any dynamics. In §3 we apply our theory developed to our particular example and in §4 we prove the existence of value for games with partial differential equations.
A local spectral theory for operators. V. Spectral subspaces for hyponormal operators
Authors: Stampfli Joseph G.
Published: July 2010

In the first part of the paper we show that the local resolvent of a hyponormal operator satisfies a rather stringent growth condition. This result enables one to show that under a mild restriction, hyponormal operators satisfy Dunford’s C condition. This in turn leads to a number of corollaries concerning invariant subspaces. In the second part we consider the local spectrum of a subnormal operator. The third section is concerned with the study of quasi-triangular hyponormal operators.
Effective lower bounds for some linear forms
Authors: Cusick T. W.
Published: July 2010

It is proved that if

1
,
α
,
β

1, \alpha ,\beta

are numbers, linearly independent over the rationals, in a real cubic number field, then given any real number

d
⩾
2

d \geqslant 2

, for any integers

x
0

,

x
1

,

x
2

{x_0},{x_1},{x_2}

such that

|

norm

(

x
0

+
α

x
1

+
β

x
2

)

|

⩽
d

|{\text {norm}}({x_0} + \alpha {x_1} + \beta {x_2})| \leqslant d

, there exist effectively computable numbers

c
>
0

c > 0

and

k
>
0

k > 0

depending only on

α
\alpha

and

β
\beta

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue x 1 x 2 EndAbsoluteValue left-parenthesis log StartAbsoluteValue x 1 x 2 EndAbsoluteValue right-parenthesis Superscript k log d Baseline StartAbsoluteValue x 0 plus alpha x 1 plus beta x 2 EndAbsoluteValue greater-than c">

|

x
1

x
2

|

(
log
⁡

|

x
1

x
2

|

)

k
log
⁡
d

|

x
0

+
α

x
1

+
β

x
2

|

>
c

|{x_1}{x_2}|{(\log |{x_1}{x_2}|)^{k\log d}}|{x_0} + \alpha {x_1} + \beta {x_2}| > c

holds whenever

x
1

x
2

≠
0

{x_1}{x_2} \ne 0

. It would be of much interest to remove the dependence on d in the exponent of

log
⁡

|

x
1

x
2

|

\log |{x_1}{x_2}|

, for then, among other things, one could deduce, for cubic irrationals, a stronger and effective form of Roth’s Theorem.
Compact nilmanifold extensions of ergodic actions
Authors: Zimmer Robert J.
Published: July 2010

We study extensions of dymanical systems defined by co-cycles into nilpotent Lie groups.
On Kolmogorov’s inequalities 𝑓̃_{𝑝}≤𝐶_{𝑝}, 𝑓₁, 0<𝑝<1
Authors: Davis Burgess
Published: July 2010

Let

μ
\mu

be a signed measure on the unit circle A of the complex plane satisfying

|

μ

|

(
A
)
>
∞

|\mu |(A) > \infty

, where

|

μ

|

(
A
)

|\mu |(A)

is the total variation of

μ
\mu

, and let

μ
~

\tilde \mu

be the conjugate function of

μ
\mu

. A theorem of Kolmogorov states that for each real number p between 0 and 1 there is an absolute constant

C
p

{C_p}

such that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis left-parenthesis 2 pi right-parenthesis Superscript negative 1 Baseline integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue ModifyingAbove mu With tilde left-parenthesis e Superscript i theta Baseline right-parenthesis EndAbsoluteValue Superscript p Baseline d theta right-parenthesis Superscript 1 slash p Baseline less-than-or-slanted-equals upper C Subscript p Baseline StartAbsoluteValue mu EndAbsoluteValue left-parenthesis upper A right-parenthesis">

(

(
2
π

)

−
1

∫
0

2
π

|

μ
~

(

e

i
θ

)

|

p

d
θ

)

1

/

p

⩽

C
p

|

μ

|

(
A
)

{({(2\pi )^{ - 1}}\smallint _0^{2\pi }|\tilde \mu ({e^{i\theta }}){|^p}d\theta )^{1/p}} \leqslant {C_p}|\mu |(A)

. Here it is shown that measures putting equal and opposite mass at points directly opposite from each other on the unit circle, and no mass any place else, are extremal for all of these inequalities, that is, if

ν
\nu

is one of these measures the number
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis left-parenthesis 2 pi right-parenthesis Superscript negative 1 Baseline integral Subscript 0 Superscript 2 pi Baseline StartAbsoluteValue ModifyingAbove nu With tilde left-parenthesis e Superscript i theta Baseline right-parenthesis EndAbsoluteValue Superscript p Baseline d theta right-parenthesis Superscript 1 slash p Baseline slash StartAbsoluteValue nu EndAbsoluteValue left-parenthesis upper A right-parenthesis">

(

(
2
π

)

−
1

∫
0

2
π

|

ν
~

(

e

i
θ

)

|

p

d
θ

)

1

/

p

/

|

ν

|

(
A
)

{({(2\pi )^{ - 1}}\smallint _0^{2\pi }|\tilde \nu ({e^{i\theta }}){|^p}d\theta )^{1/p}}/|\nu |(A)

is the smallest possible value for

C
p

{C_p}

. These constants are also the best possible in the analogous Hilbert transform inequalities. The proof is based on probability theory.
Hilbert transforms associated with plane curves
Authors: Nagel Alexander , Wainger Stephen
Published: July 2010

Let

(
t
,
γ
(
t
)
)

(t,\gamma (t))

be a plane curve. Set

H
γ

f
(
x
,
y
)
=
p.v.

∫
f
(
x
−
t
,
y
−
γ
(
t
)
)
d
t

/

t

{H_\gamma }f(x,y) = \text {p.v.}\;\smallint f(x - t,y - \gamma (t))dt/t

for

f
∈

C
0
∞

(

R
2

)

f \in C_0^\infty ({R^2})

. For a large class of curves, the authors prove
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper H Subscript gamma Baseline f double-vertical-bar Subscript p Baseline less-than-or-slanted-equals upper A Subscript p Baseline double-vertical-bar f double-vertical-bar Subscript p Baseline comma 5 slash 3 greater-than p greater-than 5 slash 2">

‖

H
γ

f

‖

p

⩽

A
p

‖
f
‖

p

,
5

/

3
>
p
>
5

/

2

{\left \| {{H_\gamma }f} \right \|_p} \leqslant {A_p}{\left \| f \right \|_p},5/3 > p > 5/2

. Various examples are given to show that some condition on the curve

(
t
,
γ
(
t
)
)

(t,\gamma (t))

is necessary.
Characterizations of continua in which connected subsets are arcwise connected
Authors: Tymchatyn E. D.
Published: July 2010

The purpose of this paper is to give several characterizations of the continua in which all connected subsets are arcwise connected. The methods used are those developed by B. Knaster and K. Kuratowski, G. T. Whyburn and the author. These methods depend on Bernstein’s decomposition of a topologically complete metric space into totally imperfect sets and on Whyburn’s theory of local cutpoints. Some properties of connected sets in finitely Suslinian spaces are obtained. Two questions raised by the author are answered. Several partial results of Whyburn are obtained as corollaries of the main result.
The module of indecomposables for finite 𝐻-spaces
Authors: Kane Richard
Published: July 2010

The module of indecomposables obtained from the

mod
p

\bmod p

cohomology of a finite H-space is studied for p odd. General structure theorems are obtained, first, regarding the possible even dimensions in which this module can be nonzero and, secondly, regarding how the Steenrod algebra acts on the module.
Smith theory for𝑝-groups
Authors: Maiorana James A.
Published: July 2010

When a p-group G acts on a manifold, the behavior of the cohomology of the subgroups of G singles out a special collection of fixed point sets of these subgroups. A bound on the size of the spaces in this collection is derived using equivariant cohomology. For a special class of nonabelian p-groups this bound is strong enough to require that certain fixed point sets must vanish. Application of this bound to a linear representation of G yields a lower bound for the cohomology of G.
On nullity distributions
Authors: Tan Sin Leng
Published: July 2010

The nullity concept of Riemannian manifolds is extended to affine manifolds. Results obtained by Chern and Kuiper and Maltz on Riemannian manifolds are generalized to affine manifolds. A structure theorem for affine symmetric spaces is obtained. Finally, the nullity concept is generalized to study the partial integrability of certain geometric structures.
Existence theorems for parametric problems in the calculus of variations and approximation
Authors: Goor Robert M.
Published: July 2010

In this paper, we investigate the parametric growth condition which arises in connection with existence theorems for parametric problems of the calculus of variations. In particular, we study conditions under which the length of a curve is dominated in a suitable sense by its “cost". We show that we may restrict our attention to local growth conditions on a particular set. Then we link the growth conditions to a certain approximation problem on this set. Finally, we prove that under suitable topological restrictions related to dimension theory, the local and global problems can be solved.
Principal co-fiber bundles
Authors: Katz Elyahu
Published: July 2010

Principal co-fiber bundles are defined in the category of topological groups. They are Eckmann-Hilton duals of principal fiber bundles. A classification theorem is provided as well as an example which almost represents the most general case.
Necessary and sufficient conditions for the derivation of integrals of 𝐿_{𝜓}-functions
Authors: Hayes C. A.
Published: July 2010

It has been shown recently that a necessary and sufficient condition for a derivation basis to derive the

μ
\mu

-integrals of all functions in

L

(
q
)

(
μ
)

{L^{(q)}}(\mu )

, where

1
>
q
>
+
∞

1 > q > + \infty

, and

μ
\mu

is a

σ
\sigma

-finite measure, is that the basis possess Vitali-like covering properties, with covering families having arbitrarily small

L

(
p
)

(
μ
)

{L^{(p)}}(\mu )

-overlap, where

p

−
1

+

q

−
1

=
1

{p^{ - 1}} + {q^{ - 1}} = 1

. The corresponding theorem for the case

p
=
1
,
q
=
+
∞

p = 1,q = + \infty

was established by R. de Possel in 1936. The present paper extends these results to more general dual Orlicz spaces. Under suitable restrictions on the dual Orlicz functions

Φ
\Phi

and

Ψ
\Psi

, it is shown that a necessary and sufficient condition for a basis to derive the

μ
\mu

-integrals of all functions in

L
Ψ

(
μ
)

{L_\Psi }(\mu )

is that the basis possess Vitali-like covering families whose

L
Φ

(
μ
)

{L_\Phi }(\mu )

-overlap is arbitrarily small. Certain other conditions relating

L
Φ

(
μ
)

{L_\Phi }(\mu )

-strength and derivability are also discussed.
Espaces fibrés linéaires faiblement négatifs sur un espace complexe
Authors: Ancona Vincenzo
Published: July 2010

Let F be a coherent sheaf over a compact reduced complex space

X
,
L
(
F
)

X,L(\text {F})

the linear fibre space associated with F,

S
k

(
F
)

{S^k}(\text {F})

the kth symmetric power of F. We show that if the zero-section of

L
(
F
)

L(\text {F})

is exceptional, then

H
r

(
X
,
E

⊗

O
X

S
k

(
F
)
)
=
0

{H^r}(X,\text {E}{ \otimes _{{O_X}}}{S^k}(\text {F})) = 0

for every coherent sheaf E on X and for

r
⩾
1

r \geqslant 1

and sufficiently large k. Using this result, we deduce moreover that Supp F is a Moišezon space.
Linear factorization of conical polynomials over certain nonassociative algebras
Authors: Lepowsky J.
Published: July 2010

Conical polynomials are defined as certain polynomials in quadratic elements of the universal enveloping algebra of a semisimple symmetric Lie algebra over a field of characteristic zero. These polynomials were used in an earlier paper to describe the conical vectors in certain induced modules. Here it is shown that when the base field is extended to a certain type of nonassociative algebra, the conical polynomials can be factored “linearly". One such nonassociative algebra is discussed in detail—an (alternative) composition algebra intimately related to the structure of the Lie algebra and studied earlier by B. Kostant in the context of real semisimple Lie algebras. The linear factorization leads in a later paper to an extension of the earlier work on conical vectors in induced modules.
Frobenius calculations of Picard groups and the Birch-Tate-Swinnerton-Dyer conjecture
Authors: Hoobler Raymond T.
Published: July 2010

Let

Y
⊂

P

m

Y \subset {{\text {P}}^m}

be a subvariety of codimension d defined by an ideal I in char

p
>
0

p > 0

with

H
1

(
Y
,

O

(
−
1
)
)
=
0

{H^1}(Y,\mathcal {O}( - 1)) = 0

. If t is an integer greater than

log
p

(
d
)

{\log _p}(d)

and

H
i

(
Y
,

I
n

/

I

n
+
1

)
=
0

{H^i}(Y,{I^n}/{I^{n + 1}}) = 0

for

n
>>
0

n > > 0

and

i
=
1
,
2

i = 1,2

, then

Pic

(
Y
)

{\text {Pic}}(Y)

is an extension of a finite p-primary group of exponent at most

p
t

{p^t}

by

Z
[

O

(
1
)
]

Z[\mathcal {O}(1)]

and

Br

′

(
Y
)
(
p
)

{\text {Br}}’(Y)(p)

is a group of exponent at most

p
t

{p^t}

. If Y is also smooth and defined over a finite field with

dim
⁡
Y
>
p

\dim Y > p

and

p
≠
2

p \ne 2

, then the B-T-SD conjecture holds for cycles of codimension 1. These results are proved by studying the etale cohomology of the Frobenius neighborhoods of Y in

P

m

{{\text {P}}^m}

.
Interposition and lattice cones of functions
Authors: Blatter Jörg , Seever G. L.
Published: July 2010

A lattice cone of functions on a set X is a convex cone of bounded real-valued functions on X which contains the constants and which is closed under the lattice operations. Our principal results concern the relation between closed lattice cones on a set X and certain binary relations, called inclusions, on the power set of X. These results are applied to interposition problems, Császár compactifications of quasi-proximity spaces, the compactification of Nachbin’s completely regular ordered topological spaces, and a problem in best approximation.
Systems-conjugate and focal points of fourth order nonselfadjoint differential equations
Authors: Cheng Sui Sun
Published: July 2010

Systems-conjugate points have been studied by John Barrett in relation to selfadjoint fourth order differential equations of the form

(

p
2

u
)
+

p
0

u
=
0

({p_2}u) + {p_0}u = 0

. This paper extends his results to the general nonselfadjoint fourth order differential equation via a system of second order equations.
Balanced subgroups of abelian groups
Authors: Hunter Roger H.
Published: July 2010

The balanced subgroups of Fuchs are generalised to arbitrary abelian groups. Projectives and injectives with respect to general balanced exact sequences are classified; a new class of groups is introduced in order to classify these projectives.
Boundary value problems for functional differential equations with 𝐿² initial functions
Authors: Reddien G. W. , Webb G. F.
Published: July 2010

Existence results are given for boundary value problems for vector systems of functional differential equations with

L
2

{L^2}

initial functions. The proofs are essentially constructive and lead to computational methods in important cases.
Strong differentiability properties of Bessel potentials
Authors: Deignan Daniel J. , Ziemer William P.
Published: July 2010

This paper is concerned with the “strong”

L
p

{L_p}

differentiability properties of Bessel potentials of order

α
>
0

\alpha > 0

of

L
p

{L_p}

functions. Thus, for such a function f, we investigate the size (in the sense of an appropriate capacity) of the set of points x for which there is a polynomial

P
x

(
y
)

{P_x}(y)

of degree

k
⩽
α

k \leqslant \alpha

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit sup Underscript diam left-parenthesis upper S right-parenthesis right-arrow 0 Endscripts left-parenthesis diam upper S right-parenthesis Superscript negative k Baseline left-brace StartAbsoluteValue upper S EndAbsoluteValue Superscript negative 1 Baseline integral StartAbsoluteValue f left-parenthesis y right-parenthesis minus upper P Subscript x Baseline left-parenthesis y right-parenthesis EndAbsoluteValue Superscript p Baseline d y right-brace Superscript 1 slash p Baseline equals 0">

lim

sup

diam

(
S
)
→
0

(

diam

S

)

−
k

{

|

S

|

−
1

∫

|

f
(
y
)
−

P
x

(
y
)

|

p

d
y

}

1

/

p

=
0

\lim \sup \limits _{{\text {diam}}(S) \to 0} \;{({\text {diam}}\;S)^{ - k}}{\left \{ {|S{|^{ - 1}}\int {|f(y) - {P_x}(y){|^p}dy} } \right \}^{1/p}} = 0

\]
where, for example, S is allowed to run through the family of all oriented rectangles containing the origin.
Spectral geometry of symmetric spaces
Authors: Gilkey Peter B.
Published: July 2010

Let M be a compact Riemannian manifold without boundary. Let D be a differential operator on M. Let spec (D, M) denote the eigenvalues of D repeated according to multiplicity. Several authors have studied the extent to which the geometry of M is reflected by spec (D, M) for certain natural operators D. We consider operators D which are convex combinations of the ordinary Laplacian and the Bochner or reduced Laplacian acting on the space of smooth functions and the space of smooth one forms. We prove that is is possible to determine if M is a local symmetric space from its spectrum. If the Ricci tensor is parallel transported, the eigenvalues of the Ricci tensor are spectral invariants of M.
The classification of stunted projective spaces by stable homotopy type
Authors: Feder S. , Gitler S.
Published: July 2010

A complete classification of stable homotopy types of complex and quaternionic stunted projective spaces, denoted by

C

P
n
k

{\mathbf {C}}P_n^k

and

Q

P
n
k

{\mathbf {Q}}P_n^k

respectively, is obtained. The necessary conditions for such equivalences are found using K-theory and various characteristic classes introduced originally by J. F. Adams. As a by-product one finds the J-orders of the Hopf bundles over

C

P
n

{\mathbf {C}}{P^n}

and

Q

P
n

{\mathbf {Q}}{P^n}

respectively. The algebraic part is rather involved. Finally a homotopy theoretical argument yields the constructions of such homotopy equivalences as are allowed by the fulfillment of the necessary conditions.
Limit theorems for convolution iterates of a probability measure on completely simple or compact semigroups
Authors: Mukherjea A.
Published: July 2010

This paper extends the study (initiated by M. Rosenblatt) of the asymptotic behavior of the convolution sequence of a probability measure on compact or completely simple semigroups. Let S be a locally compact second countable Hausdorff topological semigroup. Let

μ
\mu

be a regular probability measure on the Borel subsets of S such that S does not have a proper closed subsemigroup containing the support F of

μ
\mu

. It is shown in this paper that when S is completely simple with its usual product representation

X
×
G
×
Y

X \times G \times Y

, then the convolution sequence

μ
n

{\mu ^n}

converges to zero vaguely if and only if the group factor G is noncompact. When the group factor G is compact,

μ
n

{\mu ^n}

converges weakly if and only if

lim
_

n
→
∞

F
n

{\underline {\lim } _{n \to \infty }}{F^n}

is nonempty. This last result remains true for an arbitrary compact semigroup S generated by F. Furthermore, we show that in this case there exist elements

a
n

∈
S

{a_n} \in S

such that

μ
n

∗

δ

a
n

{\mu ^n} \ast {\delta _{{a_n}}}

converges weakly, where

δ

a
n

{\delta _{{a_n}}}

is the point mass at

a
n

{a_n}

. This result cannot be extended to the locally compact case, even when S is a group.
Lebesgue summability of double trigonometric series
Authors: Kohn M. J.
Published: July 2010

We formulate a definition of symmetric derivatives of odd order for functions of two variables. Our definition is based on expanding in a Taylor’s series a weighted average of the function about circles. The definition is applied to derive results on Lebesgue summability for spherically convergent double trigonometric series.
Balayage in Fourier transforms: general results, perturbation, and balayage with sparse frequencies
Authors: Shapiro George S.
Published: July 2010

Let

Λ
\Lambda

be a discrete subset of an LCA group and E a compact subset of the dual group. Balayage is said to be possible for

(
Λ
,
E
)

(\Lambda ,E)

if the Fourier transform of each measure on G is equal on E to the Fourier transform of some measure supported by

Λ
\Lambda

. Following Beurling, we show that this condition is equivalent to the possibility of bounding certain functions with spectra in E by their bounds on

Λ
\Lambda

. We derive consequences of this equivalence, among them a necessary condition on

Λ
\Lambda

for balayage when E is compact and open (a condition analogous to a density condition Beurling and Landau gave for balayage in Euclidean spaces). We show that if balayage is possible for

(
Λ
,
E
)

(\Lambda ,E)

and if

Λ
′

\Lambda ’

is close to

Λ
\Lambda

, then balayage is possible for

(

Λ
′

,
E
)

(\Lambda ’,E)

. Explicit bounds for the needed closeness in R and

R
n

{R^n}

are given. Using these perturbation techniques, we give examples of perfect sets

E
⊂
R

E \subset R

with the property that there are “arbitrarily sparse” sets

Λ
\Lambda

with balayage possible for

(
Λ
,
E
)

(\Lambda ,E)

.
A resolvent for an iteration method for nonlinear partial differential equations
Authors: Neuberger J. W.
Published: July 2010

For each of m and n a positive integer denote by

S
(
m
,
i
)

S(m,i)

the space of all real-valued symmetric i-linear functions on

E
m

,
i
=
1
,
2
,
…
,
n

{E_m},i = 1,2, \ldots ,n

. Denote by L a nonzero linear functional on

S
(
m
,
n
)

S(m,n)

, denote by f a real-valued analytic function on

E
m

×
R
×
S
(
m
,
1
)
×
⋯
×
S
(
m
,
[
n

/

2
]
)

{E_m} \times R \times S(m,1) \times \cdots \times S(m,[n/2])

and denote by

α
\alpha

a member of

D
(
f
)

D(f)

. Denote by H the space of all real-valued functions U, analytic at the origin of

E
m

{E_m}

, so that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha equals left-parenthesis 0 comma upper U left-parenthesis 0 right-parenthesis comma upper U prime left-parenthesis 0 right-parenthesis comma ellipsis comma upper U Superscript left-parenthesis left-bracket n slash 2 right-bracket right-parenthesis Baseline left-parenthesis 0 right-parenthesis right-parenthesis">

α
=
(
0
,
U
(
0
)
,

U
′

(
0
)
,
…
,

U

(
[
n

/

2
]
)

(
0
)
)

\alpha = (0,U(0),U’(0), \ldots ,{U^{([n/2])}}(0))

. For
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U element-of upper H comma f Subscript upper U Baseline left-parenthesis x right-parenthesis identical-to f left-parenthesis x comma upper U left-parenthesis x right-parenthesis comma upper U prime left-parenthesis x right-parenthesis comma ellipsis comma upper U Superscript left-parenthesis left-bracket n slash 2 right-bracket right-parenthesis Baseline left-parenthesis x right-parenthesis right-parenthesis">

U
∈
H
,

f
U

(
x
)
≡
f
(
x
,
U
(
x
)
,

U
′

(
x
)
,
…
,

U

(
[
n

/

2
]
)

(
x
)
)

U \in H,{f_U}(x) \equiv f(x,U(x),U’(x), \ldots ,{U^{([n/2])}}(x))

for all x for which this is defined. A one-parameter semigroup (nonlinear if

f
≠
0

f \ne 0

) K on H is constructed so that if

U
∈
K

U \in K

, then

K
(
λ
)
U

K(\lambda )U

converges, as

λ
→
∞

\lambda \to \infty

, to a solution Y to the partial differential equation

L

Y

(
n
)

=

f
Y

L{Y^{(n)}} = {f_Y}

. A resolvent j for this semigroup is determined so that

J
(
λ
)
U

J(\lambda )U

also converges to y as

λ
→
∞

\lambda \to \infty

and so that

J

(
λ

/

n

)
n

U

J{(\lambda /n)^n}U

converges to

K
(
λ
)
U

K(\lambda )U

as

n
→
∞

n \to \infty

. The solutions

Y
∈
H

Y \in H

of

L

Y

(
n
)

=

f
Y

L{Y^{(n)}} = {f_Y}

are precisely the fixed points of the semigroup K.
On subcategories of TOP
Authors: Franklin S. P. , Lutzer D. J. , Thomas B. V. S.
Published: July 2010

A categorical characterization of a subcategory S of TOP (or

T
2

{T_2}

) is one which enables the identification of S in TOP (or

T
2

{T_2}

) without requiring the reconstruction of the topological structure of its objects. In this paper we so characterize various familiar subcategories of TOP (Hausdorff spaces, normal spaces, compact Hausdorff spaces, paracompact Hausdorff spaces, metrizable spaces, first countable spaces) in terms of the global behavior of the (objects and) morphisms of the subcategory.
On the bordism of almost free 𝑍_{2𝑘} actions
Authors: Beem R. Paul
Published: July 2010

An “almost free”

Z

2
k

{Z_{{2^k}}}

action on a manifold is one in which only the included

Z
2

{Z_2}

may possibly fix points of the manifold. For k = 2, these are the stationary-point free actions. It is shown that almost free

Z

2
k

{Z_{{2^k}}}

bordism is generated by three subalgebras: the extension from

Z
2

{Z_2}

actions, a coset of

Z
2

{Z_2}

extensions being the restrictions of circle actions and a certain ideal of elements which annihilate the whole ring. The additive structure is determined. Free

Z

2
k

{Z_{{2^k}}}

bordism is shown to split as an algebra. It is shown that the kernel of the extension homomorphism from

Z
2

{Z_2}

to

Z

2
k

{Z_{{2^k}}}

bordism is equal to the image of the corresponding restriction homomorphism.
The orderability and suborderability of metrizable spaces
Authors: Purisch S.
Published: July 2010

A space is defined to be suborderable if it is embeddable in a (totally) orderable space. It is shown that a metrizable space X is suborderable iff (1) each component of X is orderable, (2) the set of cut points of each component of X is open, and (3) each closed subset of X which is a union of components has a base of clopen neighborhoods. Note that condition (1) and hence this result is topological since there are many good topological characterizations of connected orderable spaces. In a space X let Q denote the union of all nondegenerate components each of whose noncut points has no compact neighborhood. It is also shown that a metrizable space X is orderable iff (1) X is suborderable, (2)

X
−
Q

X - Q

is not a proper compact open subset of X, and (3) if W is a neighborhood of

p
∈
X

p \in X

and K is the component in X containing p such that

(
W
−
K
)
−
Q

(W - K) - Q

has compact closure and

{
p
}

\{ p\}

is the intersection of the closures of

(
W
−
K
)
−
Q

(W - K) - Q

and

(
W
−
K
)
∩
Q

(W - K) \cap Q

, then K is a singleton. Corollaries are given; every condition in each of these corollaries is concisely stated and sufficient for a space to be orderable when it is metrizable and suborderable. Both of these results are extended to a class properly containing the metrizable spaces.
Second-order differential equations with fractional transition points
Authors: Olver F. W. J.
Published: July 2010

An investigation is made of the differential equation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d squared w slash d x squared equals left-brace u squared left-parenthesis x minus x 0 right-parenthesis Superscript lamda Baseline f left-parenthesis u comma x right-parenthesis plus g left-parenthesis u comma x right-parenthesis slash left-parenthesis x minus x 0 right-parenthesis squared right-brace w comma">

d
2

w

/

d

x
2

=
{

u
2

(
x
−

x
0

)
λ

f
(
u
,
x
)
+
g
(
u
,
x
)

/

(
x
−

x
0

)
2

}
w
,

{d^2}w/d{x^2} = \{ {u^2}{(x - {x_0})^\lambda }f(u,x) + g(u,x)/{(x - {x_0})^2}\} w,

\]
in which u is a large real (or complex) parameter,

λ
\lambda

is a real constant such that

λ
>
−
2

\lambda > -2

, x is a real (or complex) variable, and

f
(
u
,
x
)

f(u,x)

and

g
(
u
,
x
)

g(u,x)

are continuous (or analytic) functions of x in a real interval (or complex domain) containing

x
0

{x_0}

. The interval (or domain) need not be bounded. Previous results of Langer and Riekstins giving approximate solutions in terms of Bessel functions of order

1

/

(
λ
+
2
)

1/(\lambda + 2)

are extended and error bounds supplied.
Essential central range and selfadjoint commutators in properly infinite von Neumann algebras
Authors: Halpern Herbert
Published: July 2010

The essential central range of an element A of a von Neumann algebra with respect to a central ideal is characterized as those elements arbitrarily close to the compression of A to a subspace large with respect to the ideal. The selfadjoint commutators in a properly infinite algebra are shown to be the elements whose essential central ranges with respect to the strong radical contain 0.
Hopf invariants and Browder’s work on the Kervaire invariant problem
Authors: Krueger Warren M.
Published: July 2010

In this paper we calculate certain functional differentials in the Adams spectral sequence converging to Wu cobordism whose values may be thought of as Hopf invariants. These results are applied to reobtain Browder’s characterization: if

q
+
1
=

2
k

q + 1 = {2^k}

, there is a 2q dimensional manifold of Kervaire invariant one if and only if

h
k
2

h_k^2

survives to

E
∞

(

S
0

)

{E_\infty }({S^0})

.
Hausdorff content and rational approximation in fractional Lipschitz norms
Authors: O’Farrell Anthony G.
Published: July 2010

For

0
>
α
>
1

0 > \alpha > 1

, we characterise those compact sets X in the plane with the property that each function in the class

lip

(
α
,
X
)

{\text {lip}}(\alpha ,X)

that is analytic at all interior points of X is the limit in

Lip

(
α
,
X
)

{\text {Lip}}(\alpha ,X)

norm of a sequence of rational functions. The characterisation is in terms of Hausdorff content.
Sums of solid 𝑛-spheres
Authors: Broussard Lois M.
Published: July 2010

We prove that the sum of two solid Antoine n-spheres

(
n
⩾
3
)

(n \geqslant 3)

by the identity on the boundary is homeomorphic to the n-sphere

S
n

{S^n}

.
Linear operators for which 𝑇*𝑇 and 𝑇+𝑇* commute. II
Authors: Campbell Stephen L. , Gellar Ralph
Published: July 2010

Let

θ
\theta

denote the set of bounded linear operators T, acting on a separable Hilbert space

K

\mathcal {K}

such that

T
∗

T

{T^\ast }T

and

T
+

T
∗

T + {T^\ast }

commute. It is shown that such operators are

G
1

{G_1}

. A complete structure theory is developed for the case when

σ
(
T
)

\sigma (T)

does not intersect the real axis. Using this structure theory, several nonhyponormal operators in

θ
\theta

with special properties are constructed.