Once a dapp is connected to Phantom, it can request user permission to send transactions on their behalf. To initiate a transaction, ensure you have the necessary parameters. For details on constructing transactions, refer to Sui developer documentation. 
const transactionParams = {
        transaction: await tx.toJSON(), // Replace with actual transaction
        address: address.toString(), // Sender address
        networkID: SupportedSuiChainIds.SuiMainnet, // or your network ID
};
To prompt Phantom to send a transaction to the network, refer to the following code snippet: 
try {
    const signature = await provider.signTransaction(params);
    return signature;
} catch (error) {
    throw new Error(error instanceof Error ? error.message : 'Failed to sign transaction');
}