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Let n be any natural number. The n-centered operator is introduced for adjointable operators on Hilbert C * -modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, n-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator T , if T is Moore-Penrose invertible and is n-centered, then its Moore-Penrose inverse is also n-centered. A Hilbert space operator T is constructed such that T is n-centered, whereas it fails to be (n + 1)-centered.2010 Mathematics Subject Classification. Primary 46L08; Secondary 47A05.

The design of a high-gain and low-profile H -plane horn antenna embedded into a large metallic platform is proposed. The antenna is composed of three parts: coaxial-to-ridged transition, wideband H -plane antenna, and dielectric lens. The coaxial-to-ridged transition makes an efficient conversion from TEM modes in a coaxial probe to the fundamental TE 10 mode in a ridged horn, and by choosing the ratio between horn aperture and length and suppressing the TE higher mode, the optimal design of the H -plane horn antenna can be obtained. Then, a tapered dielectric lens is located in the end-fire direction, which converts the TE wave into the leaky-wave mode, enhancing the peak gain. A prototype of the proposed antenna is finally fabricated and tested. The measured results are in good agreement with the simulated ones, which show that the proposed antenna has a very wide bandwidth from 2.5 to 20 GHz for the voltage standing wave ratio (VSWR) < 2.5 and exhibits a low thickness of only 8 mm (0.066λ L , λ L is the free-space wavelength at the lowest operating frequency). Good radiation pattern and high gain can be achieved over a wide frequency band.INDEX TERMS End-fire antenna, horn antenna, tapered dielectric lens, wideband, high gain.

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