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magya_bloom
magya_bloom
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A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible subspace for a subalgebra $A \in B(\mathcal{H})$. Is it any subspace subspace that isdoes not contain a proper invariant subspace under $A$, or a closed subspace that does not contain a proper invariant subspace under $A$?

A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible subspace for a subalgebra $A \in B(\mathcal{H})$. Is it any subspace that is invariant under $A$, or a closed subspace invariant under $A$?

A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible subspace for a subalgebra $A \in B(\mathcal{H})$. Is it any subspace that does not contain a proper invariant subspace under $A$, or a closed subspace that does not contain a proper invariant subspace under $A$?

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magya_bloom
magya_bloom
  • 103
  • 5

Irreducible subspaces of separable Hilbert space

A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible subspace for a subalgebra $A \in B(\mathcal{H})$. Is it any subspace that is invariant under $A$, or a closed subspace invariant under $A$?