{"id":5374,"date":"2025-07-01T06:45:57","date_gmt":"2025-07-01T06:45:57","guid":{"rendered":"https:\/\/fs.kibu.ac.ke\/?p=5374"},"modified":"2025-12-26T22:33:33","modified_gmt":"2025-12-26T22:33:33","slug":"circularity-of-numerical-ranges-for-isometrically-bounded-operators","status":"publish","type":"post","link":"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/","title":{"rendered":"Circularity of Numerical Ranges for Isometrically Bounded Operators"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"5374\" class=\"elementor elementor-5374\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-37eb98c1 elementor-section-full_width elementor-section-height-min-height elementor-section-items-stretch elementor-section-height-default\" data-id=\"37eb98c1\" data-element_type=\"section\" id=\"ourgoal\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-314ec156\" data-id=\"314ec156\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-2c14f01a elementor-widget elementor-widget-theme-post-title elementor-page-title elementor-widget-heading\" data-id=\"2c14f01a\" data-element_type=\"widget\" data-widget_type=\"theme-post-title.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Circularity of Numerical Ranges for Isometrically Bounded Operators<\/h1>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-4c71be89 elementor-section-height-min-height elementor-section-boxed elementor-section-height-default elementor-section-items-middle\" data-id=\"4c71be89\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-401d53c5\" data-id=\"401d53c5\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-3874833a elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"3874833a\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-717b4add elementor-invisible\" data-id=\"717b4add\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6cf6b347 elementor-widget__width-auto elementor-widget elementor-widget-heading\" data-id=\"6cf6b347\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">2024<\/h2>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-23a205ba elementor-invisible\" data-id=\"23a205ba\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-1f0de786 elementor-widget elementor-widget-heading\" data-id=\"1f0de786\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">KIBU Authors<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-1101728b elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"1101728b\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>Lucy Chikamai<\/p><p>Shem Aywa<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-4d12f9f4 elementor-invisible\" data-id=\"4d12f9f4\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-41271a43 elementor-tablet-align-left elementor-invisible elementor-widget elementor-widget-button\" data-id=\"41271a43\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInLeft&quot;}\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-shrink\" href=\"https:\/\/www.irejournals.com\/paper-details\/1705906\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">VIEW ON PUBLISHER SITE<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-2fef06df elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"2fef06df\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-2cf9fdc1 elementor-invisible\" data-id=\"2cf9fdc1\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-6c60d471 elementor-widget elementor-widget-heading\" data-id=\"6c60d471\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Abstract<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-77c6ead1 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"77c6ead1\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>This paper investigates the circularity of numerical ranges for isometrically bounded operators on Hilbert spaces. We establish a complete characterization of the class of isometrically bounded operators with circular numerical ranges in terms of their unitary equivalence to scalar multiples of unitary operators, which we call the Circular Numerical Range Theorem. Several corollaries and equivalent formulations of this result are derived, unifying and extending previous results on the circularity of numerical ranges for specific operator classes. Furthermore, we establish the Circular Boundary Theorem, relating the circularity of the numerical range to the essential spectrum of the operator, and the Circular Convex Hull Theorem, characterizing circular numerical ranges as the closed convex hull of the essential spectrum. These results offer new insights into the relationship between the geometry of the numerical range and the spectral properties of the operator. Throughout the paper, we provide carefully chosen examples and counterexamples to illustrate the key aspects of the theory and demonstrate the sharpness of our findings. The implications of our results for operator theory and potential applications in various fields, such as quantum mechanics and matrix analysis, are discussed. Our work contributes to the general theory of numerical ranges and opens up new avenues for research in functional analysis and related areas.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>2024 KIBU Authors Lucy Chikamai Shem Aywa VIEW ON PUBLISHER SITE Abstract This paper investigates the circularity of numerical ranges for isometrically bounded operators on Hilbert spaces. We establish a complete characterization of the class of isometrically bounded operators with circular numerical ranges in terms of their unitary equivalence to scalar multiples of unitary operators, [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_header_footer","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5374","post","type-post","status-publish","format-standard","hentry","category-research-list"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Circularity of Numerical Ranges for Isometrically Bounded Operators - Faculty of Science<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Circularity of Numerical Ranges for Isometrically Bounded Operators - Faculty of Science\" \/>\n<meta property=\"og:description\" content=\"2024 KIBU Authors Lucy Chikamai Shem Aywa VIEW ON PUBLISHER SITE Abstract This paper investigates the circularity of numerical ranges for isometrically bounded operators on Hilbert spaces. We establish a complete characterization of the class of isometrically bounded operators with circular numerical ranges in terms of their unitary equivalence to scalar multiples of unitary operators, [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/\" \/>\n<meta property=\"og:site_name\" content=\"Faculty of Science\" \/>\n<meta property=\"article:published_time\" content=\"2025-07-01T06:45:57+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-26T22:33:33+00:00\" \/>\n<meta name=\"author\" content=\"kibabii\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"kibabii\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/\"},\"author\":{\"name\":\"kibabii\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/fc9b7f13eeb65cf0be520197c35b17d6\"},\"headline\":\"Circularity of Numerical Ranges for Isometrically Bounded Operators\",\"datePublished\":\"2025-07-01T06:45:57+00:00\",\"dateModified\":\"2025-12-26T22:33:33+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/\"},\"wordCount\":228,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#organization\"},\"articleSection\":[\"Research List\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/\",\"url\":\"https:\/\/kibu.ac.ke\/fs\/circularity-of-numerical-ranges-for-isometrically-bounded-operators\/\",\"name\":\"Circularity of Numerical Ranges for Isometrically Bounded Operators - 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