On closed range operators and their characterization via p-Schatten ideals

Using the Dixmier angle between two closed subspaces of a complex Hilbert space H, we establish the necessary and sufficient conditions for the operator norm of the sum of two orthogonal projections, PW1 and PW2, onto closed subspaces W1 and W2, to attain its maximum, namely ∥PW1+PW2∥=2. These conditions are expressed in terms of the geometric relationship and symmetry between the ranges of the projections. We apply these results to orthogonal projections associated with a closed-range operator via its Moore–Penrose inverse. Additionally, for any bounded operator T with closed range in H, we derive sufficient conditions ensuring ∥TT†+T†T∥=2, where T† denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges and their algebraic structure governs norm extremality and extends a recent finite-dimensional result to the general Hilbert space setting.

Fully Nonlinear Programming Problems with Closed Range Operators

In this article, we introduce the concept of pre-quasi norm on E (Orlicz sequence space), which is more general than the usual norm, and give the conditions on E equipped with the pre-quasi norm to be Banach space. We give the necessity and sufficient conditions on E equipped with the pre-quasi norm such that the multiplication operator defined on E is a bounded, approximable, invertible, Fredholm, and closed range operator. The components of pre-quasi operator ideal formed by the sequence of s-numbers and E is strictly contained for different Orlicz functions are determined. Furthermore, we give the sufficient conditions on E equipped with a pre-modular such that the pre-quasi Banach operator ideal constructed by s-numbers and E is simple and its components are closed. Finally the pre-quasi operator ideal formed by the sequence of s-numbers and E is strictly contained in the class of all bounded linear operators, whose sequence of eigenvalues belongs to E.

Electronically steerable antennas are expensive, whilst mechanically scanned antennas are of complex construction and hence are prone to failure. Based on the leaky-wave frequency-scanning principle, the spinning grating antenna (SGA) provides beam steering without the need of hinges. Whilst pulsed or stepped frequency waveforms may be implemented with this antenna, a frequency modulated continuous wave (FMCW) waveform allows close range operation at high speed, the possibility of excellent range-resolution, and low-cost signal processing. However, to implement FMCW with the SGA, the inherent scan-angle variation with frequency must first be overcome. Following on from an effort to develop a highly accurate radar system employing closed-loop linearisation to achieve 4 GHz of bandwidth centred at 94 GHz, this paper focuses on simultaneously achieving high angular resolution over a 20 degree scan range with good results.

Izumino has discussed a sequence of closed range operators $(T_n)$ that converges to a closed range operator $T$ on a Hilbert space to establish the convergence of $T^{dag}_n$ $to$ $T^{dag}$ for Moore-Penrose inverses. In general, if $T_n to T$ uniformly and each $T_n$ has a closed range, then $T$ need not have a closed range. Some sufficient conditions have been discussed on $T_n$ and $T$ such that $T$ has a closed range whenever each $T_n$ has a closed range.

A characterization of the class of partial isometries

Let A, B and be closed range operators. The explicit matrix expressions for various generalized inverses are obtained by using block operator matrix methods. Some subtle relationships between the properties of sub-blocks in operator matrices A, B and their range relations are built. New necessary and sufficient conditions for the equivalent relations, inclusion relations and mixed-type generalized inverses relations are presented. Some recent mixed-type reverse-order laws results are covered and many new mixed-type generalized inverses relations are established by using this block-operator matrix technique.

MARESye: A hybrid imaging system for underwater robotic applications

A note on perturbations of fusion frames

A novel guidance scheme for close range operation in active debris removal

This survey aims to give a brief introduction to Wold-type decomposition for some closed range operators satisfying some operator inequalities. As a cornerstone in the theory of the Hardy space, Beurling theorem for unweighted shift is our starting point that we try to transfer to regular operators. Also, several results on left invertible operators close to isometries, as extensions of the Hardy shifts, are listed and extended to the case of regular operators. We define and study the Cauchy dual for such operators by using the Moore-Penrose inverse of closed range operators. The Cauchy dual plays the role of the left inverse in our approach for this general setting.KeywordsWold-type decompositionBeurling-type theoremRegular operatorMoore-Penrose inverseCauchy dual

Necessary conditions for the optimality of a pair \((\overline{y}, \overline{u})\) with respect to a locally Lipschitz cost functional L(y,u) , subject to Ay + F(y) = Cu + B(u) , are given in terms of generalized gradients. Here A and C are densely defined, closed, linear operators on some Banach spaces, while F and B are (Frechet) differentiable maps, which are suitably related to A and C . Various examples and potential applications to nonlinear programming models and nonlinear optimal control of partial differential equations are also discussed.

With the rapid development of nuclear medicine, the number of nuclear medical staff has increased a lot in the past few years in China. Close-range operations, such as preparation and injections of radiopharmaceuticals, are usually carried out in nuclear medicine department. And the use of unsealed radionuclides may also create internal exposure risk. So, occupational exposure of nuclear medical staff is a main issue of occupational health management in China. In this paper, the occupational exposure level and requirements for radiation protection of nuclear medical staff are introduced to provide references for the related work that radiological health technical institutions carry out.

A mm-wave imager has been developed for medical applications and will be reported here. It uses a single-channel mechanically scanned heterodyne radiometer and diffraction-limited focussing optics to achieve high resolution images of subcutaneous body temperature. The particular design constraints imposed by close-range operation (tens of centimetres) will be discussed. The instrument uses a novel, patented method for regular calibration of both the thermal and spatial response of the imager. Results obtained from healthy volunteers will also be presented.

Indefinite least-squares problems and pseudo-regularity