Some generalized theorems onp-hyponormal operators

A bounded linear operatorT is calledp-Hyponormal if (T ^*T)^p(TT ^*)^p, 0<p1. In Aluthge [1], we studied the properties of p-hyponormal operators using the operator . In this work we consider a more general operator , and generalize some properties of p-hyponormal operators obtained in [1].     

Berger-Shaw's theorem forp-hyponormal operators

For an-multicyclicp-hyponormal operatorT, we shall show that |T|^2p –|T ^*|^2p belongs to the Schatten and that tr Area ((T)).     

On realizations related to Weyl operators

The paper deals with the minimal and the maximal realizations (L ^w )^~ and (L ^w ):L ₂L ₂ of linear operators of Weyl type

Inequalities of Putnam and Berger-Shaw forp-quasihyponormal operators

For an operatorT satisfying thatT ^*(T ^* T–TT ^*)T0, we shall show that and, moreover, tr itT isn-multicyclic.For an operatorT satisfying thatT ^* {(T ^* T) ^p –(TT ^*) ^p }T0 for somep (0, 1], we shall show that and, moreover, ifT isn-multicyclic.     

A kernel for operators with one-dimensional self-commutators

It is known that if T is an operator with self-commutator of the form T^* T-TT^*=φ?φ, then for every complex z there is a unique solution x=T _z ^*?1 φ of (T?z)^*x=φ which is orthogonal to the kernel of (T-z)^*. The exponential representation $$1 - (T_z^{*^{ - 1} } \varphi ,T_w^{*^{ - 1} } \varphi ) = \exp \left\{ { - \frac{1}{\pi } \int {\mathbb{C} \frac{{g_T ^{(\zeta )} }}{{\overline {(\zeta - z)} (\zeta - w)}} \frac{{d\overline \zeta \Lambda d\zeta }}{{2i}}} } \right\}$$ where g_T is the principal function of T is established. The kernel \(\overline Q (z,w) \equiv (T_z^{*^{ - 1} } \varphi ,T_w^{*^{ - 1} } \varphi )\) has the advantage over previous kernels in that it is defined on all of ?². Several consequences of the exponential representation of Q are derived. For example, if the planar measure of the essential spectrum of T is zero, then the Cowen-Douglas curvature can be immediately computed from Q.     

Spectral analysis of finite convolution operators with matrix kernels

Let N be a finite dimensional complex Hilbert space. A finite convolution operator on the vector function space L _N ² (0,1) is an operator T of the form (Tf) (x) = ₀ ^x k (x–t)f(t)dt, where k(t) is a norm integrableB(N)-valued function on [0,1]. A symbol for T is any function A(z) of the form A(z) = ₀ ¹ k(t)e^itzdt + e^izG(z), where G(z) is aB(N)-valued function which is bounded and analytic in a half plane y > . It is shown that under suitable restrictions two finite convolution operators are similar if their symbols are asymptotically close as z in a half plane y > .This paper was written at the University of Virginia and is based on the author's doctoral dissertation [6], which was supervised by James Rovnyak.     

Trace formulas for pair operators

In this paper we shall study the Fredholm determinant and related trace formulas for a class of operators which correspond to the restriction of integral operators with kernels of the form k(x,y) = (x)g^v(x–y)+[1–(x)]f^v(x–y) to the square |x|,|y| T and shall evaluate the limit as T . Here denotes the indicator function of the right half-line [0,) . The results obtained generalize the well known formulas of M. Kac for the classical convolution operator in which g = f .     

On log-hyponormal operators

LetTB(H) be a bounded linear operator on a complex Hilbert spaceH.TB(H) is called a log-hyponormal operator itT is invertible and log (TT ^*)log (T ^* T). Since log: (0, )(–,) is operator monotone, for 0<p1, every invertiblep-hyponormal operatorT, i.e., (TT ^*) ^p (T ^* T) ^p , is log-hyponormal. LetT be a log-hyponormal operator with a polar decompositionT=U|T|. In this paper, we show that the Aluthge transform is . Moreover, ifmeas ((T))=0, thenT is normal. Also, we make a log-hyponormal operator which is notp-hyponormal for any 0<p.This research was supported by Grant-in-Aid Research No. 10640185     

On spectra ofp-hyponormal operators

The purpose of this paper is to show the following: Let 0<p<1/2. IfT=U|T| is a p-hyponormal operator with a unitaryU on a Hilbert space, then $$\sigma (T) = \mathop \cup \limits_{0 \leqslant k \leqslant 1} \sigma (T_{\left[ k \right]} ),$$ where $$T_{\left[ k \right]} = U[(1 - k)S_U^ - (\left| T \right|^{2p} ) + kS_U^ + (\left| T \right|^{2p} ]^{\tfrac{1}{{2p}}} $$ andS _U ^± (T) denote the polar symbols ofT.     

A band limited and Besov class functional calculus for Tadmor-Ritt operators

For Banach space operators T satisfying the Tadmor-Ritt condition a band limited H^∞ calculus is established, where and a is at most of the order C(T)⁵. It follows that such a T allows a bounded Besov algebra B_{∞ 1}⁰ functional calculus, These estimates are sharp in a convenient sense. Relevant embedding theorems for B_{∞ 1}⁰ are derived. Received: 25 October 2004; revised: 31 January 2005     

Toeplitz operators with noncommuting symbols

LetM be a von Neumann algebra with a faithful normal tracial state and letH be a finite maximal subdiagonal subalgebra ofM. LetH ² be the closure ofH in the noncommutative Lebesgue spaceL ²(M). We consider Toeplitz operators onH ² whose symbol belong toM, and find that they possess several of the properties of Toeplitz operators onH ²( ) with symbol fromL ( ), including norm estimates, a Hartman-Wintner spectral inclusion theorem, and a characterisation of the weak^* continuous linear functionals on the space of Toeplitz operators.     

Compact Hankel operators with conjugate analytic symbols

In 1986 S. Axler [3] proved that forf∈L _a ² the Hankel operator is compact if and only iff is in the little Bloch space {itB}{in0}. In this note we show that the same is true for , 1<p<∞. Moreover we prove that is ⋆-compact if and only if as |z|→1⁻.     

Measures on spaces of operators and isometries

One considers a question that arises at the investigation of isometric operators in vector-valued L^p -spaces. Let E, F be Banach spaces, let p>0, let μ be a probability Borel measure on the space on continuous linear operators from E into F such that for any e ε E one has $$\left\| e \right\|^p = \int {\left\| {Te} \right\|^p } d\mu \left( T \right)$$ . In the cases when: 1) E=C(K), K is a metric compactum, F is an arbitrary space, p>1 and 2) 2)E=F=L^q,p>1, q>1 q ?[p,2] it is Proved that the support of the measure μ is contained in the set of the operators that are scalar multiples of isometries. For E=C(K) one obtains an isomorphic analogue of this result: if the Banach-Mazur distance between C(K) and the P -sum of Banach spaces is small, then the distance between C(K) and one of the spaces is small.     

On the type of spectral operators and the non-spectrality of several differential operators on Lp

A spectral operator, not necessarily bounded, which is the infinitesimal generator of a strongly continuous group of operators {U(t)|t} where U(t) = (|t|^K), is of type k N_o, i.e. T = S + N, where S is spectral of scalar type, N is bounded, N^k+1 = O and S and N are commuting. This result yields a simple proof of the non-spectrality of certain differential operators on L^p, p 2, which are known to be selfadjoint for p = 2.     

Scott Brown's techniques for perturbations of decomposable operators

Using Scott Brown's techniques, J. Eschmeier and B. Prunaru showed that if T is the restriction of a decomposable (or S-decomposable) operator B to an invariant subspace such that (T) is dominating in C/S for some closed set S, then T has an invariant subspace. In the present paper we prove various invariant subspace theorems by weakening the decomposability condition on B and strengthening the thickness condition on (T).The research is supported by a grant from the Institute for Studies in Theoretical Physics and Mathematics (IRAN).     

Interpolation theorem for linear operators

In generalizing a series of known results, the following theorem is proved: If K is a continuous linear operator mapping E₀ into F₀ and E₁ into F₁ (where E₀, E₁ and F₀, F₁, being ideal spaces, are Banach lattices of functions defined on ₁ and ₂ respectively), then for any . (0, 1) K maps E ₀ ^1– E ₁ into [(F ₀ )^1–(F ₁ )] and is continuous; for suitably chosen norms in the spacesE ₀ ^1– E ₁ and [(F ₀ )^1–(F ₁ )] the norm of K is a logarithmically convex function of . Six titles are cited in the bibliography.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 593–598, December, 1967.The author wishes to thank M. A. Krasnosel'skii, under whose direction he is working.     

Adjoints of slant Toeplitz operators II

Let be the unit circle {z|z|=1} and _n c _n e ⁱⁿ be a bounded measurable function on . Theslant Toeplitz operator A onL ² ( ) is defined by A e _n ,e _m =c _2m–n for allm, n wheree _n (z)=z ⁿ , . In this paper, we continue the study initiated in [6] onA ^* , the adjoint ofA . Specifically, we will show that for a certain dense set of continuous functions on ,A ^* is similar to some constant multiple of either a shift, or a shift plus a rank one operator.     

On the modulus of disjointness preserving operators on complex vector lattices

Let L and M be Archimedean vector lattices such that and are complex vector lattices. We constructively and intrinsically prove that if is an order bounded disjointness preserving operator from into then the modulus

Generalized Hankel operators on the Fock space

In this paper we study generalized Hankel operators ofthe form : ?²(|z |²) → L²(|z |²). Here, (f):= (Id–P_l )( ^kf) and Pl is the projection onto A_l ²(?, |z |²):= cl(span{ ^m zⁿ | m, n ∈ N, m ≤ l }). The investigations in this article extend the ones in [11] and [6], where the special cases l = 0 and l = 1 are considered, respectively. The main result is that the operators are not bounded for l < k – 1. The proof relies on a combinatoric argument and a generalization to general conjugate holomorphic L² symbols, generalizing arguments from [6], seems possible and is planned for future work (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Saturation of Kantorovich type operators

It is proved that an integrable functionf can be approximated by the Kantorovich type modification of the Szász—Mirakjan and Baskakov operators inL ¹ metric in the optimal order {n ^–1} if and only if ² f is of bounded variation where and , respectively.