Vali-e-Asr university of Rafsanjan Wavelet and Linear Algebra 2383-1936 2 1 2015 09 01 Ultra Bessel sequences in direct sums of Hilbert spaces 55 64 14270 EN M. R. Abdollahpour University of Mohaghegh Ardabili A. Rahimi University of Maragheh Journal Article 2015 09 07 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">In this paper, we establish some new results in ultra Bessel sequences and ultra Bessel sequences of subspaces. Also, we investigate ultra Bessel sequences in direct sums of Hilbert spaces. <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">Specially, we show that <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">{<span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">( <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>g</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">)<span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">}<span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∞ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: rtxr; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">=<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">is a an ultra Bessel sequence <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">for Hilbert space <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">H ⊕ K <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">if and only if <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">{ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>f</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">}<span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∞ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: rtxr; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">=<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">{<span style="font-family: NimbusRomNo9L-ReguItal; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;"><em>g</em><span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">}<span style="font-family: txsy; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">∞ <span style="font-family: NimbusRomNo9L-ReguItal; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>i</em><span style="font-family: rtxr; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">=<span style="font-family: NimbusRomNo9L-Regu; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;">1 <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">are ultra <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">Bessel sequences for Hilbert spaces <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">H <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">and <span style="font-family: txsy; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">K<span style="font-family: rtxmi; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">, <span style="font-family: NimbusRomNo9L-Regu; font-size: 11pt; color: #000000; font-style: normal; font-variant: normal;">respectively.</span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> https://wala.vru.ac.ir/article_14270_60d22149eee1175d3675575416f123b1.pdf