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  "body": "\n## Summary\n\nThe resolvent set of an element $a$ of a unital algebra is the set of scalars $\\lambda$ for which $a - \\lambda 1$ is invertible.\nThe same notion applies to linear operators.\nIt is the complement of the spectrum.\nOn it the resolvent $\\lambda \\mapsto (a - \\lambda 1)^{-1}$ is defined.\nIn a complex Banach algebra the resolvent set is open, and the resolvent is holomorphic on it.\nThese properties of the resolvent are the main tools to study the spectrum.\n\n## Explanations\n\n### L1 human\n\nThe resolvent set collects the numbers $\\lambda$ for which the equation $(a - \\lambda) x = y$ can always be solved in a unique and stable way.\nFor a matrix, these are all numbers that are not eigenvalues.\nFor multiplication by the function $t$ on $[0,1]$, these are the complex numbers outside the interval $[0,1]$.\n",
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