/pdf/operators-with-finite-ascent-41xowyfu67.pdf

Operators with finite ascent.

Joint numerical ranges of operators in semi-Hilbertian spaces

If f is any vector-valued bounded function defined on open set D of the complex plane, and T is any bounded linear operator on a Hilbert space such that aR(T*) is empty and if (T zI)2f(z) = x for all z in D then f is analytic. Let T be a bounded linear operator on a Hilbert space H. Let f be an H-valued function defined on an open set D of the complex plane. Suppose (T zI)f(z) = x for all z in D and for a fixed x in H. The question arises: What type of conditions on the operator T and on the functionf will be sufficient to insure thatf is analytic on D? This question has been implicitly discussed in Clancey [1], Johnson [4], Putnam [5], [6], Radjabalipour [7], and Stampfli-Wadhwa [9], [10], under various conditions related to normality on T. One of the main results is: If T is a hyponormal operator (TT* < T* T) and f is any function satisfying the above equation then f is analytic in D. See [1]. On the other hand, there is a cohyponormal operator T, (T* is hyponormal) and a bounded function f satisfying the above equation which fails to be analytic on D. See [6]. In this note, by a simple argument, we shall show that if (T zI)2f(z) = x for all z in D and if f is bounded on D and aJR(T*) = 0 (range of T* zI is dense in H for all z) thenf is analytic. We shall use this result to give an alternative proof of a classical result of Stampfli [8] about quadratically hyponormal operators. The following lemma is implicitly contained in most of the references mentioned previously. We include it for the sake of completeness. LEMMA. Let (T zI)f(z) = x for all z in D be such that f is bounded and aR(T*) = 0 then f is weakly continuous. PROOF. For any z and z0 in D, (f(z) f(zo), (T* i01)y) = ((T z0)(f(z) f(z0)), y) = (z zo)(f(z), y) for ally in H. Sincef is bounded and range of (T* i0I) is dense in H, it follows thatf is weakly continuous. THEOREM. Let g be a bounded vector-valuedfunction such that (T zI)2g(z) = x for all z in D and let aJR(T*) = 0. Then g is analytic on D. PROOF. Let f(z) = (T zI)g(z). Then (T zI)f(x) = x for all z in D, f is bounded and hence weakly continuous on each bounded subset Do of D. Now for Received by the editors October 30, 1979 and, in revised form, January 29, 1980. A MSq (MOS") subiect iagsifications (1970). Primarv 47A05. 47B20. i 1980 American Mathematical Society 0002-9939/80/0000-0625/$01.50 653 This content downloaded from 157.55.39.104 on Mon, 20 Jun 2016 06:19:40 UTC All use subject to http://about.jstor.org/terms

https://www.ams.org/proc/1980-080-04/S0002-9939-1980-0587947-8/S0002-9939-1980-0587947-8.pdf

On operator ranges

An extension of Putnam-Fuglede theorem for hyponormal operators

The intertwining relations between cosubnormal and closed hyponormal (resp. cohyponormal and closed subnormal) operators are studied. In particular, an asymmetric Putnam–Fuglede theorem for unbounded operators is proved.

/pdf/an-asymmetric-putnam-fuglede-theorem-for-unbounded-operators-2raqw9siso.pdf

An asymmetric Putnam–Fuglede theorem for unbounded operators

This paper is devoted to the study of dominant operators with an emphasis on their spectral properties. In particular the equation (T−λ)f(λ)≡ ≡x (T a dominant or hyponormal operator on the Hilbert space ℌ,x∈ℌ andf a function from the open setU to ℌ) is investigated in an effort to discover necessary and/or sufficient conditions for the analyticity off.

On dominant operators

The joint numerical status of commuting bounded operators Ai and A2 on a Hubert space is defined as {{φiA^y φ(A2)) such that φ is a state on the C*-algebra generated by Ax and A2}. It is shown that if At and A2 are commuting normal operators then their joint numerical status equals the closure of their joint numerical range. It is also shown that certain points in the boundary of the joint numerical range are joint approximate reducing eigenvalues. The joint numerical range of Ax and A2 denoted by w(Alf A2) is jX, x), (A2x, x)) such that xeH and ||g|| = 1}. Thus w(Alf A2) is a bounded subset of C 2. It is not known whether this set is convex, Dash [4, 6]. In this note, we shall show that there is faithful * representation of the C*-algebra generated by Ax and Ai9 C*(Aif A2), under which the joint numerical range of At and A2 is convex. Following Berberian and Orland [1], we study the joint numerical status of Ax and A2, Σ(Aλ, A2) = {{φ(Ax)9 Φ(A2)) such that φ is a state on C*(A19 A2)}. If Ax and A2 are commuting normal operators then Σ(Aif A2) — w(A19 A2). We also show that certain points in the boundary of w(A19 A2) are joint approximate reducing eigenvalues. For the sake of notational convenience, all the results are being stated for two commuting operators. However, the results hold for any finite family of commuting operators. Let B(H) denote the algebra of all bounded linear operators on

/pdf/on-joint-numerical-ranges-1g3y2vv13r.pdf

On joint numerical ranges

There exist in the literature many notions of jointspectra which have been then generalized to joint essentialspectra and in some instances to joint Browder spectra.The purpose of this note is to develop a notion of jointBrowder spectrum which uses ideas of Arveson and Wael-brook. It is shown that this notion of joint Browderspectrum has many of the useful properties of the Browderspectrum.Polynomial joint Browder spectrum* For an operator (boundedoperator) A on a Banach space

/pdf/joint-browder-spectrum-4c0wcjjbf3.pdf

Joint Browder spectrum

Compact-like operators and the baire category theorem

COVID has posed several challenges for higher education. There is a rise in blended teaching and learning models that can improve the quality of education, observed Raman. Classroom interaction vital in quality of education is affected by power distance between the teacher and students, states Kasuya. Much research has been done on blended learning but hardly any on the role of power distance in blended learning in higher education, specifically post COVID. Keeping in mind the growing significance of blended learning shortly, it becomes strategically important to understand the role of power distance in blended learning in higher education post-COVID-19. This research paper tries to address the research gap. The study is exploratory, exploring the role of power distancing in blended learning formats during post-COVID-19. The findings indicate that there is a relationship between power distance and the autonomy of an individual. The higher the power distance between a teacher and a student from the teacher’s perspective, it becomes teacher-centred learning. So blended learning with the best of low power distances in terms of the media and delivery mechanism of offline and online ensures the mitigation of power between the teacher and the taught. The research will help the universities create a perfect blended learning format that enhances the quality of higher education. Due to the paucity of time and resources and a ban on respondent interaction due to pandemics, this research is based on secondary data analysis.

Bhushan L. Wadhwa

Papers

On dominant operators

On joint numerical ranges

Joint Browder spectrum

Compact-like operators and the baire category theorem

Role of power distance phenomena in blended learning in higher education post-Covid-19