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Agentic artificial intelligence systems—those capable of planning, executing, adapting, and learning across operational tasks—are beginning to influence healthcare administrative workflows. Their emergence raises important questions about autonomy, safety, oversight, and accountability in hospital systems. This systematic review examined the development and deployment of agentic AI in healthcare operations from 2017 to 2026. The review focused on autonomous discharge coordination, task routing, human oversight, safety guardrails, and workflow accountability. A PRISMA 2020–aligned search was conducted across PubMed, Scopus, IEEE Xplore, and Web of Science. Studies were screened by two reviewers, and eligible records were synthesised narratively according to task type, agent capability, oversight model, safety mechanism, and deployment maturity. The evidence base was nascent, heterogeneous, and dominated by predictive, prototype, implementation, and early deployment studies. Most systems supported discharge planning, workflow prediction, task prioritisation, or operational decision support rather than fully autonomous execution. Agentic AI has substantial potential to improve hospital operations, especially in discharge coordination and workflow routing. However, real-world deployment requires stronger safety engineering, prospective evaluation, explicit human oversight, and auditable accountability structures.