[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"skill:01M6NVC8EBFP5ARAVHBFPAGA9R":3,"skill-same:01M6NVC8EBFP5ARAVHBFPAGA9R":59},{"skillNo":4,"name":5,"nameZh":6,"description":7,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":16,"categories":17,"keywords":18,"publishedAt":30,"linkOnly":31,"markdownContent":32,"markdownBody":33,"frontMatter":34,"refName":37,"commitSha":38,"skillDirectory":39,"forksCount":14,"repositoryHtmlUrl":40,"repositoryHomepageUrl":6,"ownerHtmlUrl":41,"ownerBlogUrl":42,"ownerDescription":43,"updatedAt":44,"install":45,"resources":58},"01M6NVC8EBFP5ARAVHBFPAGA9R","operator-theory",null,"Problem-solving strategies for operator theory in functional analysis","rubicanjr\u002FFinCognis","rubicanjr","https:\u002F\u002Favatars.githubusercontent.com\u002Fu\u002F234912519?v=4","01M6NT156NND8J474G34SB3923","Rubi","01M6NT15B70RWXE1N3YV0BZY8A",0,"MIT","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fmath\u002Ffunctional-analysis\u002Foperator-theory\u002FSKILL.md",[],[19,20,21,22,23,24,25,26,27,28,29],"operator","operator theory","analysis","Bash","read","problem-solving","strategies","problem","functional","for","functional analysis","2026-11-10 10:12:30",false,"---\nname: operator-theory\ndescription: \"Problem-solving strategies for operator theory in functional analysis\"\nallowed-tools: [Bash, Read]\n---\n\n# Operator Theory\n\n## When to Use\n\nUse this skill when working on operator-theory problems in functional analysis.\n\n## Decision Tree\n\n\n1. **Bounded operator verification**\n   - ||Tx|| \u003C= M||x|| for some M\n   - Operator norm: ||T|| = sup{||Tx|| : ||x|| = 1}\n   - `z3_solve.py prove \"operator_bounded\"`\n\n2. **Adjoint operator**\n   - \u003CTx, y> = \u003Cx, T*y> defines T*\n   - For matrices: T* = conjugate transpose\n   - `sympy_compute.py simplify \"\u003CTx, y> - \u003Cx, T*y>\"`\n\n3. **Spectral Theory**\n   - Spectrum: sigma(T) = {lambda : T - lambda*I not invertible}\n   - Self-adjoint: spectrum is real\n   - `z3_solve.py prove \"self_adjoint_real_spectrum\"`\n\n4. **Compact operators**\n   - T compact if T(bounded set) has compact closure\n   - Approximable by finite-rank operators\n   - `sympy_compute.py limit \"||T - T_n||\" --var n`\n\n5. **Spectral Theorem**\n   - Self-adjoint compact: T = sum(lambda_n * P_n)\n   - eigenvalues -> 0, eigenvectors form orthonormal basis\n\n\n## Tool Commands\n\n### Z3_Bounded_Operator\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"norm(Tx) \u003C= M*norm(x)\"\n```\n\n### Sympy_Adjoint\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py simplify \"\u003CTx, y> - \u003Cx, T_star_y>\"\n```\n\n### Z3_Spectral\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"self_adjoint implies real_spectrum\"\n```\n\n### Sympy_Compact\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py limit \"norm(T - T_n)\" --var n --at oo\n```\n\n## Key Techniques\n\n*From indexed textbooks:*\n\n- [Introductory Functional Analysis with Applications] Spectral theory is one of the main branches of modern functional analysis and its applications. Roughly speaking, it is concerned with certain inverse operators, their general properties and their relations to the original operators. Such inverse operators arise quite naturally in connection with the problem of solving equations (systems of linear algebraic equations, differential equations, integral equations).\n- [Introductory Functional Analysis with Applications] Unbounded linear operators in Hilb,ert spaces will be considered in Chap. Brief orientation about main content of Chap. We begin with finite dimensional vector spaces.\n- [Introductory Functional Analysis with Applications] Most unbounded linear operators occurring in practical problems are closed or have closed linear extensions (Sec. Unbounded Linear Operators in Hilbert Space The spectrum of a self-adjoint linear operator is real, also in the unbounded case (d. T is obtained by means of the Cayley transform U= (T- iI)(T+ iI)-1 of T (d.\n- [Introductory Functional Analysis with Applications] Compact Operators and Their Spectrum is called a degenerate kernel. Here we may assume each of the two sets {ab· . If an equation (1) with such a kernel has a solution x, show that it must be of the form n x(s' = ji(s) + lot L cjaj(s), j~l and the unknown constants must satisfy cj - n lot L ajkCk = Yj' k~l where j= 1,···, n.\n- [Introductory Functional Analysis with Applications] As indicated before, our key to the application of complex analysis to spectral theory will be Theorem 7. The theorem states that for every value AoEp(n the resolvent R>. TE B(X, X) on a complex Banach space X has a power series repre- sentation (4) R>.\n\n## Cognitive Tools Reference\n\nSee `.claude\u002Fskills\u002Fmath-mode\u002FSKILL.md` for full tool documentation.\n","# Operator Theory\n\n## When to Use\n\nUse this skill when working on operator-theory problems in functional analysis.\n\n## Decision Tree\n\n\n1. **Bounded operator verification**\n   - ||Tx|| \u003C= M||x|| for some M\n   - Operator norm: ||T|| = sup{||Tx|| : ||x|| = 1}\n   - `z3_solve.py prove \"operator_bounded\"`\n\n2. **Adjoint operator**\n   - \u003CTx, y> = \u003Cx, T*y> defines T*\n   - For matrices: T* = conjugate transpose\n   - `sympy_compute.py simplify \"\u003CTx, y> - \u003Cx, T*y>\"`\n\n3. **Spectral Theory**\n   - Spectrum: sigma(T) = {lambda : T - lambda*I not invertible}\n   - Self-adjoint: spectrum is real\n   - `z3_solve.py prove \"self_adjoint_real_spectrum\"`\n\n4. **Compact operators**\n   - T compact if T(bounded set) has compact closure\n   - Approximable by finite-rank operators\n   - `sympy_compute.py limit \"||T - T_n||\" --var n`\n\n5. **Spectral Theorem**\n   - Self-adjoint compact: T = sum(lambda_n * P_n)\n   - eigenvalues -> 0, eigenvectors form orthonormal basis\n\n\n## Tool Commands\n\n### Z3_Bounded_Operator\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"norm(Tx) \u003C= M*norm(x)\"\n```\n\n### Sympy_Adjoint\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py simplify \"\u003CTx, y> - \u003Cx, T_star_y>\"\n```\n\n### Z3_Spectral\n```bash\nuv run python -m runtime.harness scripts\u002Fz3_solve.py prove \"self_adjoint implies real_spectrum\"\n```\n\n### Sympy_Compact\n```bash\nuv run python -m runtime.harness scripts\u002Fsympy_compute.py limit \"norm(T - T_n)\" --var n --at oo\n```\n\n## Key Techniques\n\n*From indexed textbooks:*\n\n- [Introductory Functional Analysis with Applications] Spectral theory is one of the main branches of modern functional analysis and its applications. Roughly speaking, it is concerned with certain inverse operators, their general properties and their relations to the original operators. Such inverse operators arise quite naturally in connection with the problem of solving equations (systems of linear algebraic equations, differential equations, integral equations).\n- [Introductory Functional Analysis with Applications] Unbounded linear operators in Hilb,ert spaces will be considered in Chap. Brief orientation about main content of Chap. We begin with finite dimensional vector spaces.\n- [Introductory Functional Analysis with Applications] Most unbounded linear operators occurring in practical problems are closed or have closed linear extensions (Sec. Unbounded Linear Operators in Hilbert Space The spectrum of a self-adjoint linear operator is real, also in the unbounded case (d. T is obtained by means of the Cayley transform U= (T- iI)(T+ iI)-1 of T (d.\n- [Introductory Functional Analysis with Applications] Compact Operators and Their Spectrum is called a degenerate kernel. Here we may assume each of the two sets {ab· . If an equation (1) with such a kernel has a solution x, show that it must be of the form n x(s' = ji(s) + lot L cjaj(s), j~l and the unknown constants must satisfy cj - n lot L ajkCk = Yj' k~l where j= 1,···, n.\n- [Introductory Functional Analysis with Applications] As indicated before, our key to the application of complex analysis to spectral theory will be Theorem 7. The theorem states that for every value AoEp(n the resolvent R>. TE B(X, X) on a complex Banach space X has a power series repre- sentation (4) R>.\n\n## Cognitive Tools Reference\n\nSee `.claude\u002Fskills\u002Fmath-mode\u002FSKILL.md` for full tool documentation.\n",{"name":5,"description":7,"allowed-tools":35},[22,36],"Read","main","70e1c350500ee750987420bbae48837afa906b7e","skills\u002Fmath\u002Ffunctional-analysis\u002Foperator-theory","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis","https:\u002F\u002Fgithub.com\u002Frubicanjr","linktr.ee\u002Ffinanskodu","Industrial Engineer & CIO @FinansKodu. Architecting AI-driven financial agents and autonomous workflows. 🏗️ Building systems, not just prompts.","2026-06-09 13:50:15",{"source":8,"skillName":5,"commands":46,"prompt":56,"zipAvailable":57},[47,50,53],{"tool":48,"command":49},"npx","npx skills add rubicanjr\u002FFinCognis --skill operator-theory",{"tool":51,"command":52},"bunx","bunx skills add rubicanjr\u002FFinCognis --skill operator-theory",{"tool":54,"command":55},"pnpm","pnpm dlx skills add rubicanjr\u002FFinCognis --skill operator-theory","安装技能，技能地址：https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fmath\u002Ffunctional-analysis\u002Foperator-theory\u002FSKILL.md",true,[],[60,81,90,109,138,157],{"skillNo":61,"name":62,"nameZh":6,"description":63,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":64,"categories":65,"keywords":71,"publishedAt":30},"01M6NWS9DAXJTADT4T5MB49X8W","workflow-router","Goal-based workflow orchestration - routes tasks to specialist agents based on user goals","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fworkflow-router\u002FSKILL.md",[66],{"categoryNo":67,"categoryName":68,"taxonomyType":69,"taxonomyTypeDesc":70},"01JBSEEDCAT000000000000023","Skill与智能体开发","SCENARIO","使用场景",[72,73,74,75,76,77,78,79,80],"workflow router","workflow","router","orchestration","goals","goal","user","workflow orchestration","tasks",{"skillNo":82,"name":83,"nameZh":6,"description":84,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":85,"categories":86,"keywords":87,"publishedAt":30},"01M6NWS3RETSC0SHP16HP66E14","wiring","Wiring Verification","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fwiring\u002FSKILL.md",[],[83,88,89],"verification","false",{"skillNo":91,"name":92,"nameZh":6,"description":93,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":94,"categories":95,"keywords":99,"publishedAt":30},"01M6NWS0XW9CVQZBDA1EGW0TWC","websocket-patterns","Connection management, room patterns, reconnection strategies, message buffering, and binary protocol design.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fwebsocket-patterns\u002Fskill.md",[96],{"categoryNo":97,"categoryName":98,"taxonomyType":69,"taxonomyTypeDesc":70},"01JBSEEDCAT000000000000015","视觉设计",[100,101,102,103,104,105,106,25,107,108],"websocket","patterns","websocket patterns","design","protocol","management","message","binary","connection",{"skillNo":110,"name":111,"nameZh":6,"description":112,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":113,"categories":114,"keywords":119,"publishedAt":30},"01M6NWRDQDX9ZNETAWRRMQWV52","visual-verdict","Screenshot comparison QA for frontend development. Takes a screenshot of the current implementation, scores it across multiple visual dimensions, and returns a structured PASS\u002FREVISE\u002FFAIL verdict with concrete fixes. Use when implementing UI from a design reference or verifying visual correctness.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fvisual-verdict\u002FSKILL.md",[115,118],{"categoryNo":116,"categoryName":117,"taxonomyType":69,"taxonomyTypeDesc":70},"01JBSEEDCAT000000000000009","前端开发",{"categoryNo":67,"categoryName":68,"taxonomyType":69,"taxonomyTypeDesc":70},[120,121,122,123,124,125,126,127,128,28,129,130,131,103,132,133,134,135,136,137],"visual","verdict","visual verdict","ui","across","revise","comparison","correctness","implementing","fixes","dimensions","frontend","development","qa","screenshot","returns","use","frontend development",{"skillNo":139,"name":140,"nameZh":6,"description":141,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":142,"categories":143,"keywords":144,"publishedAt":30},"01M6NWR8HSNRS7MNDK2J02QYMN","verification-loop","Comprehensive verification system covering build, types, lint, tests, security, and diff review before a PR.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fverification-loop\u002FSKILL.md",[],[145,146,88,147,148,149,150,151,152,153,154,155,156],"verification loop","loop","lint","review","security","pr","system","diff","build","tests","diff review","types",{"skillNo":158,"name":159,"nameZh":6,"description":160,"descriptionZh":6,"repositoryFullName":8,"ownerLogin":9,"ownerAvatarUrl":10,"ownerAccountNo":11,"ownerDisplayName":12,"repositoryNo":13,"starsCount":14,"licenseSpdxId":15,"licenseFileUrl":6,"sourceHtmlUrl":161,"categories":162,"keywords":163,"publishedAt":30},"01M6NWR4HHSFR0VR5EXFRWA747","vector-db-patterns","Embedding strategies, ANN algorithms, hybrid search, RAG chunking strategies, and reranking for semantic search and retrieval.","https:\u002F\u002Fgithub.com\u002Frubicanjr\u002FFinCognis\u002Fblob\u002Fmain\u002Fskills\u002Fvector-db-patterns\u002Fskill.md",[],[164,165,166,167,101,168,169,170,171,25,172,173,174,175,176,28,177,178],"vector db patterns","vector db","db","vector","semantic search","search","embedding strategies","algorithms","chunking","semantic","retrieval","embedding","Hybrid Search","hybrid","rag"]